TOPICS
SURROUNDING
THE
COMBINATORIAL
ANABELIAN
GEOMETRY
OF
HYPERBOLIC
CURVES
II:
TRIPODS
AND
COMBINATORIAL
CUSPIDALIZATION
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
OCTOBER
2021
Abstract.
Let
Σ
be
a
subset
of
the
set
of
prime
numbers
which
is
either
equal
to
the
entire
set
of
prime
numbers
or
of
cardinality
one.
In
the
present
monograph,
we
continue
our
study
of
the
pro-Σ
fundamental
groups
of
hyperbolic
curves
and
their
associated
con-
figuration
spaces
over
algebraically
closed
fields
in
which
the
primes
of
Σ
are
invertible.
The
starting
point
of
the
theory
of
the
present
monograph
is
a
combinatorial
anabelian
result
which,
unlike
results
obtained
in
previous
papers,
allows
one
to
eliminate
the
hypothesis
that
cuspidal
inertia
subgroups
are
preserved
by
the
isomorphism
in
question.
This
result
allows
us
to
[partially]
generalize
combinato-
rial
cuspidalization
results
obtained
in
previous
papers
to
the
case
of
outer
automorphisms
of
pro-Σ
fundamental
groups
of
configuration
spaces
that
do
not
necessarily
preserve
the
cuspidal
inertia
subgroups
of
the
various
one-dimensional
subquotients
of
such
a
fundamental
group.
Such
partial
combinatorial
cuspidalization
results
allow
one
in
effect
to
reduce
issues
concerning
the
anabelian
geometry
of
con-
figuration
spaces
to
issues
concerning
the
anabelian
geometry
of
hyperbolic
curves.
These
results
also
allow
us,
in
the
case
of
config-
uration
spaces
of
sufficiently
large
dimension,
to
give
purely
group-
theoretic
characterizations
of
the
cuspidal
inertia
subgroups
of
the
various
one-dimensional
subquotients
of
the
pro-Σ
fundamental
group
of
a
configuration
space.
We
then
turn
to
the
study
of
tripod
synchronization,
i.e.,
roughly
speaking,
the
phenomenon
that
an
outer
automorphism
of
the
pro-Σ
fundamental
group
of
a
log
config-
uration
space
associated
to
a
stable
log
curve
typically
induces
the
same
outer
automorphism
on
the
various
subquotients
of
such
a
fun-
damental
group
determined
by
tripods
[i.e.,
copies
of
the
projective
line
minus
three
points].
Our
study
of
tripod
synchronization
allows
us
to
show
that
outer
automorphisms
of
pro-Σ
fundamental
groups
of
configuration
spaces
exhibit
somewhat
different
behavior
from
the
behavior
that
may
be
observed
—
as
a
consequence
of
the
classical
Dehn-Nielsen-Baer
theorem
—
in
the
case
of
discrete
fundamen-
tal
groups.
Other
applications
of
the
theory
of
tripod
synchronization
include
a
result
concerning
commuting
profinite
Dehn
multi-
twists
that,
a
priori,
arise
from
distinct
semi-graphs
of
anabelioids
2010
Mathematics
Subject
Classification.
Primary
14H30;
Secondary
14H10.
Key
words
and
phrases.
anabelian
geometry,
combinatorial
anabelian
geometry,
combinatorial
cuspidalization,
profinite
Dehn
twist,
tripod,
tripod
synchronization,
Grothendieck-Teichmüller
group,
semi-graph
of
anabelioids,
hyperbolic
curve,
con-
figuration
space.
The
first
author
was
supported
by
Grant-in-Aid
for
Scientific
Research
(C),
No.
24540016,
Japan
Society
for
the
Promotion
of
Science.
1
2
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
of
pro-Σ
PSC-type
structures
[i.e.,
the
profinite
analogue
of
the
notion
of
a
decomposition
of
a
hyperbolic
topological
surface
into
hyperbolic
subsurfaces,
such
as
“pants”],
as
well
as
the
computation,
in
terms
of
a
certain
scheme-theoretic
fundamental
group,
of
the
purely
combinatorial/group-theoretic
commensurator
of
the
group
of
profi-
nite
Dehn
multi-twists.
Finally,
we
show
that
the
condition
that
an
outer
automorphism
of
the
pro-Σ
fundamental
group
of
a
stable
log
curve
lift
to
an
outer
automorphism
of
the
pro-Σ
fundamental
group
of
the
corresponding
n-th
log
configuration
space,
where
n
≥
2
is
an
integer,
is
compatible,
in
a
suitable
sense,
with
localization
on
the
dual
graph
of
the
stable
log
curve.
This
localizability
prop-
erty,
together
with
the
theory
of
tripod
synchronization,
is
applied
to
construct
a
purely
combinatorial
analogue
of
the
natural
outer
surjection
from
the
étale
fundamental
group
of
the
moduli
stack
of
hyperbolic
curves
over
Q
to
the
absolute
Galois
group
of
Q.
Contents
Introduction
Notations
and
Conventions
1.
Combinatorial
anabelian
geometry
in
the
absence
of
group-theoretic
cuspidality
2.
Partial
combinatorial
cuspidalization
for
F-admissible
outomorphisms
3.
Synchronization
of
tripods
4.
Glueability
of
combinatorial
cuspidalizations
References
2
15
18
32
51
103
167
Introduction
Let
Σ
⊆
Primes
be
a
subset
of
the
set
of
prime
numbers
Primes
which
is
either
equal
to
Primes
or
of
cardinality
one.
In
the
present
monograph,
we
continue
our
study
of
the
pro-Σ
fundamental
groups
of
hyperbolic
curves
and
their
associated
configuration
spaces
over
al-
gebraically
closed
fields
in
which
the
primes
of
Σ
are
invertible
[cf.
[MzTa],
[CmbCsp],
[NodNon],
[CbTpI]].
Before
proceeding,
we
review
some
fundamental
notions
that
play
a
central
role
in
the
present
monograph.
We
shall
say
that
a
scheme
X
over
an
algebraically
closed
field
k
is
a
semi-stable
curve
if
X
is
connected
and
proper
over
k,
and,
moreover,
for
each
closed
point
x
of
X,
the
completion
of
the
local
ring
O
X,x
is
isomorphic
over
k
either
to
k[[t]]
or
to
k[[t
1
,
t
2
]]/(t
1
t
2
),
where
t,
t
1
,
and
t
2
are
indeterminates.
We
shall
say
that
a
scheme
X
over
a
scheme
S
is
a
semi-stable
curve
if
the
structure
morphism
X
→
S
is
flat,
and,
moreover,
every
geometric
fiber
of
X
→
S
is
a
semi-stable
curve.
We
shall
say
that
a
pair
(X,
D)
consisting
of
a
scheme
X
over
a
scheme
S
and
a
[possibly
empty]
closed
subscheme
D
⊆
X
is
a
pointed
stable
curve
over
S
if
the
following
COMBINATORIAL
ANABELIAN
TOPICS
II
3
conditions
are
satisfied:
X
is
a
semi-stable
curve
over
S;
D
is
contained
in
the
smooth
locus
of
the
structure
morphism
X
→
S
and
étale
over
S;
the
invertible
sheaf
ω
X/S
(D)
—
where
we
write
ω
X/S
for
the
dualizing
sheaf
of
X/S
—
is
relatively
ample
[relative
to
the
morphism
X
→
S].
We
shall
say
that
a
scheme
X
over
a
scheme
S
is
a
hyperbolic
curve
over
S
if
there
exists
a
pointed
stable
curve
(Y,
E)
over
S
such
that
Y
is
smooth
over
S,
and,
moreover,
X
is
isomorphic
to
Y
\
E
over
S.
It
is
well-known
[cf.
[SGA1],
Exposé
V,
§7]
that
if
X
is
a
connected
locally
noetherian
scheme,
and
x
→
X
is
a
geometric
point
of
X,
then
the
category
Fét(X)
consisting
of
X-schemes
Z
whose
structure
morphism
is
finite
and
étale
and
[necessarily
finite
étale]
X-morphisms
forms
a
Galois
category,
for
which
the
functor
from
Fét(X)
to
the
cat-
egory
of
finite
sets
given
by
Z
→
Z
×
X
x
is
a
fundamental
functor
[cf.
[SGA1],
Exposé
V,
Définition
5.1].
Thus,
it
follows
from
the
general
theory
of
Galois
categories
[cf.
the
discussion
following
[SGA1],
Ex-
posé
V,
Remarque
5.10]
that
one
may
associate,
to
the
Galois
category
Fét(X)
equipped
with
the
above
fundamental
functor,
the
“fundamen-
tal
pro-group”
of
the
Galois
category
Fét(X)
equipped
with
the
above
fundamental
functor,
which
we
shall
refer
to
as
the
étale
fundamental
group
π
1
(X,
x)
of
(X,
x).
If
X
is
a
normal
scheme,
K
is
an
algebraic
closure
of
the
function
field
K
of
X,
and
x
is
the
tautological
geometric
point
of
X
determined
by
K,
then
π
1
(X,
x)
may
be
naturally
identi-
fied
with
the
quotient
of
Gal(K/K)
determined
by
the
union
of
finite
subextensions
K
⊆
L
⊆
K
such
that
the
normalization
of
X
in
L
is
finite
étale
over
X
[cf.
[SGA1],
Exposé
I,
Corollaire
10.3].
Since
[one
verifies
easily
that]
the
étale
fundamental
group
is,
in
a
natural
sense,
independent,
up
to
inner
automorphism,
of
the
choice
of
the
basepoint,
i.e.,
the
geometric
point
“x”,
we
shall
omit
mention
of
the
basepoint
throughout
the
present
monograph.
Let
G
be
a
topological
group.
Then
we
shall
write
Aut(G)
for
the
group
of
[continuous]
automorphisms
of
G,
Inn(G)
⊆
Aut(G)
for
the
def
group
of
inner
automorphisms
of
G,
and
Out(G)
=
Aut(G)/Inn(G)
for
the
group
of
[continuous]
outomorphisms
[i.e.,
outer
automorphisms]
of
G.
Thus,
an
outer
automorphism
of
G
is
an
automorphism
of
G
considered
up
to
composition
with
an
inner
automorphism.
Let
k
be
a
field,
k
sep
a
separable
closure
of
k,
and
X
a
geometrically
def
connected
scheme
of
finite
type
over
k.
Write
G
k
=
Gal(k
sep
/k)
for
the
absolute
Galois
group
of
k.
Then
it
is
well-known
[cf.
[SGA1],
Exposé
IX,
Théorème
6.1]
that
the
natural
morphisms
of
schemes
X
×
k
k
sep
→
X
→
Spec
k
determine
an
exact
sequence
of
profinite
groups
1
−→
π
1
(X
×
k
k
sep
)
−→
π
1
(X)
−→
G
k
−→
1.
Write
Δ
X
for
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
π
1
(X
×
k
k
sep
)
of
X
×
k
k
sep
and
Π
X
for
the
quotient
of
the
étale
fundamental
group
π
1
(X)
of
X
by
the
normal
closed
subgroup
of
π
1
(X)
4
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
determined
by
the
kernel
of
the
natural
surjection
(π
1
(X)
←)
π
1
(X
×
k
k
sep
)
Δ
X
.
Then
the
above
displayed
exact
sequence
determines
an
exact
sequence
of
profinite
groups
1
−→
Δ
X
−→
Π
X
−→
G
k
−→
1.
Next,
observe
that
the
above
displayed
exact
sequence
induces
a
natural
action
of
Π
X
on
Δ
X
by
conjugation,
i.e.,
a
homomorphism
Π
X
→
Aut(Δ
X
),
which
restricts
to
the
tautological
homomorphism
Δ
X
→
Inn(Δ
X
).
Thus,
by
considering
the
respective
quotients
by
Δ
X
,
we
obtain
an
outer
action
of
G
k
on
Δ
X
,
i.e.,
a
homomorphism
G
k
−→
Out(Δ
X
).
This
outer
action
is
one
of
the
main
objects
of
study
in
anabelian
geometry.
In
the
situation
of
the
preceding
paragraph,
if
X
is
a
hyperbolic
curve
over
k,
then
each
cusp
of
X
[i.e.,
each
geometric
point
of
the
smooth
compactification
of
X
whose
image
is
not
contained
in
X]
determines
a
conjugacy
class
of
closed
subgroups
of
Δ
X
[i.e.,
the
inertia
subgroup(s)
associated
to
the
cusp],
each
member
of
which
we
shall
refer
to
as
a
cuspidal
inertia
subgroup
of
Δ
X
.
Now
suppose
further
that
k
is
the
field
of
fractions
of
a
complete
regular
local
ring
R,
and
that
every
element
of
Σ
is
invertible
in
R.
Suppose,
moreover,
that
X
has
a
stable
model
over
R,
i.e.,
that
there
exists
a
pointed
stable
curve
(Y,
E)
over
def
S
=
Spec
R
such
that
X
is
isomorphic
to
(Y
\
E)
×
R
k
over
k.
Then
combinatorial
anabelian
geometry
may
be
described
as
the
study
of
the
combinatorial
geometric
properties
of
the
irreducible
components
and
nodes
[i.e.,
singular
points]
of
the
geometric
fiber
of
(Y,
E)
over
the
unique
closed
point
of
S
by
means
of
the
purely
group-theoretic
properties
of
the
outer
action
of
G
k
—
or,
alternatively,
various
natural
subquotients
of
G
k
—
on
Δ
X
.
Here,
we
observe
that
this
geometric
fiber
of
(Y,
E)
over
the
unique
closed
point
of
S
may
be
regarded
as
a
sort
of
degeneration
of
the
hyperbolic
curve
X.
Let
k
be
an
algebraically
closed
field
of
characteristic
∈
Σ
and
X
a
hyperbolic
curve
over
k.
For
each
positive
integer
m,
write
•
X
m
for
the
m-th
configuration
space
of
X,
i.e.,
the
open
sub-
scheme
of
the
fiber
product
of
m
copies
of
X
over
k
obtained
by
removing
the
various
diagonals;
•
Π
m
for
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
π
1
(X
m
)
of
X
m
;
def
def
•
X
0
=
Spec
k
and
Π
0
=
{1}.
Let
n
be
a
positive
integer.
We
shall
think
of
the
factors
of
X
n
as
labeled
by
the
indices
1,
.
.
.
,
n.
Thus,
for
E
⊆
{1,
.
.
.
,
n}
a
subset
of
cardinality
n
−
m,
where
m
is
a
nonnegative
integer,
we
have
a
projection
morphism
X
n
→
X
m
obtained
by
forgetting
the
factors
that
belong
to
E,
hence
also
an
induced
outer
surjection
Π
n
Π
m
,
i.e.,
a
COMBINATORIAL
ANABELIAN
TOPICS
II
5
surjection
considered
up
to
composition
with
an
inner
automorphism.
Normal
closed
subgroups
Ker(Π
n
Π
m
)
⊆
Π
n
obtained
in
this
way
will
be
referred
to
as
fiber
subgroups
of
Π
n
of
length
n
−
m
[cf.
[MzTa],
Definition
2.3,
(iii)].
Write
X
n
−→
X
n−1
−→
.
.
.
−→
X
m
−→
.
.
.
−→
X
1
−→
X
0
for
the
projections
obtained
by
forgetting,
successively,
the
factors
la-
beled
by
indices
>
m
[as
m
ranges
over
the
nonnegative
integers
≤
n].
Thus,
we
obtain
a
sequence
of
outer
surjections
Π
n
Π
n−1
.
.
.
Π
m
.
.
.
Π
1
Π
0
.
def
For
each
nonnegative
integer
m
≤
n,
write
K
m
=
Ker(Π
n
Π
m
).
Thus,
we
have
a
filtration
of
subgroups
{1}
=
K
n
⊆
K
n−1
⊆
.
.
.
⊆
K
m
⊆
.
.
.
⊆
K
1
⊆
K
0
=
Π
n
.
In
the
situation
of
the
previous
paragraph,
let
Y
be
a
hyperbolic
curve
over
k
and
Y
n
a
positive
integer.
Write
Y
Π
Y
n
for
the
“Π
n
”
that
occurs
in
the
case
where
we
take
“(X,
n)”
to
be
(Y,
Y
n).
Let
∼
α
:
Π
n
→
Y
Π
Y
n
be
a(n)
[continuous]
outer
isomorphism.
Then
we
shall
say
that
•
α
is
PF-admissible
[cf.
[CbTpI],
Definition
1.4,
(i)]
if
α
induces
a
bijection
between
the
set
of
fiber
subgroups
of
Π
n
and
the
set
of
fiber
subgroups
of
Y
Π
Y
n
;
•
α
is
PC-admissible
[cf.
[CbTpI],
Definition
1.4,
(ii),
as
well
as
Lemma
3.2,
(i),
of
the
present
monograph]
if,
for
each
positive
integer
a
≤
n,
α(K
a
)
⊆
Y
Π
Y
n
is
a
fiber
subgroup
of
Y
Π
Y
n
of
length
Y
n
−
a,
and,
moreover,
the
Y
Π
Y
n
-conjugacy-orbit
of
∼
isomorphisms
K
a−1
/K
a
→
α(K
a−1
)/α(K
a
)
determined
by
α
induces
a
bijection
between
the
set
of
conjugacy
classes
of
cus-
pidal
inertia
subgroups
of
K
a−1
/K
a
and
the
set
of
conjugacy
classes
of
cuspidal
inertia
subgroups
of
α(K
a−1
)/α(K
a
)
[where
we
note
that
it
follows
immediately
from
the
various
defini-
tions
involved
that
the
profinite
group
K
a−1
/K
a
(respectively,.
α(K
a−1
)/α(K
a
))
is
equipped
with
a
natural
structure
of
pro-Σ
surface
group
—
cf.
[MzTa],
Definition
1.2];
•
α
is
PFC-admissible
[cf.
[CbTpI],
Definition
1.4,
(iii)]
if
α
is
PF-admissible
and
PC-admissible.
Suppose,
moreover,
that
(X,
n)
=
(Y,
Y
n),
which
thus
implies
that
α
is
a(n)
[continuous]
outomorphism
of
Π
n
=
Y
Π
Y
n
.
Then
we
shall
say
that
•
α
is
F-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
if
α(K)
=
K
for
every
fiber
subgroup
K
of
Π
n
;
•
α
is
C-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
if
α
is
PC-admissible,
and
α(K
a
)
=
K
a
for
each
nonnegative
integer
a
≤
n;
6
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
•
α
is
FC-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
if
α
is
F-admissible
and
C-admissible.
One
central
theme
of
the
present
monograph
is
the
issue
of
n-
cuspidalizability
[cf.
Definition
3.20],
i.e.,
the
issue
of
the
extent
to
which
a
given
isomorphism
between
the
pro-Σ
fundamental
groups
of
a
pair
of
hyperbolic
curves
lifts
[necessarily
uniquely,
up
to
a
per-
mutation
of
factors
—
cf.
[NodNon],
Theorem
B]
to
a
PFC-admissible
[cf.
[CbTpI],
Definition
1.4,
(iii)]
isomorphism
between
the
pro-Σ
fun-
damental
groups
of
the
corresponding
n-th
configuration
spaces,
for
n
≥
1
a
positive
integer.
In
this
context,
we
recall
that
both
the
alge-
braic
and
the
anabelian
geometry
of
such
configuration
spaces
revolves
around
the
behavior
of
the
various
diagonals
that
are
removed
from
direct
products
of
copies
of
the
given
curve
in
order
to
construct
these
configuration
spaces.
From
this
point
of
view,
it
is
perhaps
natural
to
think
of
the
issue
of
n-cuspidalizability
as
a
sort
of
abstract
profinite
analogue
of
the
notion
of
n-differentiability
in
the
theory
of
differen-
tial
manifolds.
In
particular,
it
is
perhaps
natural
to
think
of
the
theory
of
the
present
monograph
[as
well
as
of
[MzTa],
[CmbCsp],
[NodNon],
[CbTpI]]
as
a
sort
of
abstract
profinite
analogue
of
the
classical
theory
constituted
by
the
differential
topology
of
surfaces.
Next,
we
recall
that,
to
a
substantial
extent,
the
theory
of
combina-
torial
cuspidalization
[i.e.,
the
issue
of
n-cuspidalizability]
developed
in
[CmbCsp]
may
be
thought
of
as
an
essentially
formal
consequence
of
the
combinatorial
anabelian
result
obtained
in
[CmbGC],
Corol-
lary
2.7,
(iii).
In
a
similar
vein,
the
generalization
of
this
theory
of
[CmbCsp]
that
is
summarized
in
[NodNon],
Theorem
B,
may
be
re-
garded
as
an
essentially
formal
consequence
of
the
combinatorial
an-
abelian
result
given
in
[NodNon],
Theorem
A.
The
development
of
the
theory
of
the
present
monograph
follows
this
pattern
to
a
substantial
extent.
That
is
to
say,
in
§1,
we
begin
the
development
of
the
the-
ory
of
the
present
monograph
by
proving
a
fundamental
combinatorial
anabelian
result
[cf.
Theorem
1.9],
which
generalizes
the
combinato-
rial
anabelian
results
given
in
[CmbGC],
Corollary
2.7,
(iii);
[NodNon],
Theorem
A.
A
substantial
portion
of
the
main
results
obtained
in
the
remainder
of
the
present
monograph
may
be
understood
as
consisting
of
various
applications
of
Theorem
1.9.
By
comparison
to
the
combinatorial
anabelian
results
of
[CmbGC],
Corollary
2.7,
(iii);
[NodNon],
Theorem
A,
the
main
technical
feature
of
the
combinatorial
anabelian
result
given
in
Theorem
1.9
of
the
present
monograph
is
that
it
allows
one,
to
a
substantial
extent,
to
eliminate
the
group-theoretic
cuspidality
hypothesis
—
i.e.,
the
assumption
to
the
effect
that
the
isomorphism
between
pro-
Σ
fundamental
groups
of
stable
log
curves
under
consideration
[that
is
to
say,
in
effect,
an
isomorphism
between
the
pro-Σ
fundamental
groups
COMBINATORIAL
ANABELIAN
TOPICS
II
7
of
certain
degenerations
of
hyperbolic
curves]
necessarily
preserves
cus-
pidal
inertia
subgroups
—
that
plays
a
central
role
in
the
proofs
of
ear-
lier
combinatorial
anabelian
results.
In
§2,
we
apply
Theorem
1.9
to
obtain
the
following
[partial]
combinatorial
cuspidalization
result
[cf.
Theorem
2.3,
(i),
(ii);
Corollary
3.22],
which
[partially]
generalizes
[NodNon],
Theorem
B.
Theorem
A
(Partial
combinatorial
cuspidalization
for
F-ad-
missible
outomorphisms).
Let
(g,
r)
be
a
pair
of
nonnegative
inte-
gers
such
that
2g
−
2
+
r
>
0;
n
a
positive
integer;
Σ
a
set
of
prime
numbers
which
is
either
equal
to
the
set
of
all
prime
numbers
or
of
car-
dinality
one;
X
a
hyperbolic
curve
of
type
(g,
r)
over
an
algebraically
closed
field
of
characteristic
∈
Σ;
X
n
the
n-th
configuration
space
of
X;
Π
n
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
X
n
;
Out
F
(Π
n
)
⊆
Out(Π
n
)
the
subgroup
of
F-admissible
outomorphisms
[i.e.,
roughly
speaking,
outer
automorphisms
that
preserve
the
fiber
subgroups
—
cf.
the
dis-
cussion
preceding
Theorem
A;
[CmbCsp],
Definition
1.1,
(ii),
for
more
details]
of
Π
n
;
Out
FC
(Π
n
)
⊆
Out
F
(Π
n
)
the
subgroup
of
FC-admissible
outomorphisms
[i.e.,
roughly
speak-
ing,
outer
automorphisms
that
preserve
the
fiber
subgroups
and
the
cuspidal
inertia
subgroups
—
cf.
the
discussion
preceding
Theorem
A;
[CmbCsp],
Definition
1.1,
(ii),
for
more
details]
of
Π
n
.
Then
the
fol-
lowing
hold:
(i)
Write
def
n
inj
=
1
2
if
r
=
0,
if
r
=
0
,
def
n
bij
=
3
4
if
r
=
0,
if
r
=
0
.
If
n
≥
n
inj
(respectively,
n
≥
n
bij
),
then
the
natural
homomor-
phism
Out
F
(Π
n+1
)
−→
Out
F
(Π
n
)
induced
by
the
projections
X
n+1
→
X
n
obtained
by
forgetting
any
one
of
the
n
+
1
factors
of
X
n+1
[cf.
[CbTpI],
Theorem
A,
(i)]
is
injective
(respectively,
bijective).
(ii)
Write
⎧
⎨
2
def
3
n
FC
=
⎩
4
if
(g,
r)
=
(0,
3),
if
(g,
r)
=
(0,
3)
and
r
=
0,
if
r
=
0
.
If
n
≥
n
FC
,
then
it
holds
that
Out
FC
(Π
n
)
=
Out
F
(Π
n
)
.
8
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(iii)
Suppose
that
(g,
r)
∈
{(0,
3);
(1,
1)}.
Then
the
natural
injec-
tion
[cf.
[NodNon],
Theorem
B]
Out
FC
(Π
2
)
→
Out
FC
(Π
1
)
induced
by
the
projections
X
2
→
X
1
obtained
by
forgetting
ei-
ther
of
the
two
factors
of
X
2
is
not
surjective.
Here,
we
remark
that
the
non-surjectivity
discussed
in
Theorem
A,
(iii),
is,
in
fact,
obtained
as
a
consequence
of
the
theory
of
tripod
syn-
chronization
developed
in
§3
[cf.
the
discussion
preceding
Theorem
C
below].
This
non-surjectivity
is
remarkable
in
that
it
yields
an
impor-
tant
example
of
substantially
different
behavior
in
the
theory
of
profi-
nite
fundamental
groups
of
hyperbolic
curves
from
the
corresponding
theory
in
the
discrete
case.
That
is
to
say,
in
the
case
of
the
classical
discrete
fundamental
group
of
a
hyperbolic
topological
surface,
the
sur-
jectivity
of
the
corresponding
homomorphism
may
be
derived
as
an
essentially
formal
consequence
of
the
well-known
Dehn-Nielsen-Baer
theorem
in
the
theory
of
topological
surfaces
[cf.
the
discussion
of
Re-
mark
3.22.1,
(i)].
In
particular,
it
constitutes
an
important
“counterex-
ample”
to
the
“line
of
reasoning”
[i.e.,
for
instance,
of
the
sort
which
appears
in
the
final
paragraph
of
[Lch],
§1;
the
discussion
between
[Lch],
Theorem
5.1,
and
[Lch],
Conjecture
5.2]
that
one
should
expect
essentially
analogous
behavior
in
the
theory
of
profinite
fundamental
groups
of
hyperbolic
curves
to
the
relatively
well
understood
behav-
ior
observed
classically
in
the
theory
of
discrete
fundamental
groups
of
topological
surfaces
[cf.
the
discussion
of
Remark
3.22.1,
(iii)].
Theorem
A
leads
naturally
to
the
following
strengthening
of
the
result
obtained
in
[CbTpI],
Theorem
A,
(ii),
concerning
the
group-
theoreticity
of
the
cuspidal
inertia
subgroups
of
the
various
one-
dimensional
subquotients
of
a
configuration
space
group
[cf.
Corol-
lary
2.4].
Theorem
B
(PFC-admissibility
of
outomorphisms).
In
the
no-
tation
of
Theorem
A,
write
Out
PF
(Π
n
)
⊆
Out(Π
n
)
for
the
subgroup
of
PF-admissible
outomorphisms
[i.e.,
roughly
speak-
ing,
outer
automorphisms
that
preserve
the
fiber
subgroups
up
to
a
pos-
sible
permutation
of
the
factors
—
cf.
the
discussion
preceding
Theo-
rem
A;
[CbTpI],
Definition
1.4,
(i),
for
more
details]
and
Out
PFC
(Π
n
)
⊆
Out
PF
(Π
n
)
for
the
subgroup
of
PFC-admissible
outomorphisms
[i.e.,
roughly
speak-
ing,
outer
automorphisms
that
preserve
the
fiber
subgroups
and
the
cus-
pidal
inertia
subgroups
up
to
a
possible
permutation
of
the
factors
—
cf.
the
discussion
preceding
Theorem
A;
[CbTpI],
Definition
1.4,
(iii),
COMBINATORIAL
ANABELIAN
TOPICS
II
9
for
more
details].
Let
us
regard
the
symmetric
group
on
n
letters
S
n
as
a
subgroup
of
Out(Π
n
)
via
the
natural
inclusion
S
n
→
Out(Π
n
)
obtained
by
permuting
the
various
factors
of
X
n
.
Finally,
suppose
that
(g,
r)
∈
{(0,
3);
(1,
1)}.
Then
the
following
hold:
(i)
We
have
an
equality
Out(Π
n
)
=
Out
PF
(Π
n
).
If,
moreover,
(r,
n)
=
(0,
2),
then
we
have
equalities
Out(Π
n
)
=
Out
PF
(Π
n
)
=
Out
F
(Π
n
)
×
S
n
.
(ii)
If
either
r
>
0,
n
≥
3
or
n
≥
4,
then
we
have
equalities
Out(Π
n
)
=
Out
PFC
(Π
n
)
=
Out
FC
(Π
n
)
×
S
n
.
The
partial
combinatorial
cuspidalization
of
Theorem
A
has
natural
applications
to
the
relative
and
[semi-]absolute
anabelian
geom-
etry
of
configuration
spaces
[cf.
Corollaries
2.5,
2.6],
which
gen-
eralize
the
theory
of
[AbsTpI],
§1.
Roughly
speaking,
these
results
allow
one,
in
a
wide
variety
of
cases,
to
reduce
issues
concerning
the
relative
and
[semi-]absolute
anabelian
geometry
of
configuration
spaces
to
the
corresponding
issues
concerning
the
relative
and
[semi-]absolute
anabelian
geometry
of
hyperbolic
curves.
Also,
we
remark
that
in
this
context,
we
obtain
a
purely
scheme-theoretic
result
[cf.
Lemma
2.7]
that
states,
roughly
speaking,
that
the
theory
of
isomorphisms
[of
schemes!]
between
configuration
spaces
associated
to
hyperbolic
curves
may
be
reduced
to
the
theory
of
isomorphisms
[of
schemes!]
between
hyper-
bolic
curves.
In
§3,
we
take
up
the
study
of
[the
group-theoretic
versions
of]
the
various
tripods
[i.e.,
copies
of
the
projective
line
minus
three
points]
that
occur
in
the
various
one-dimensional
fibers
of
the
log
configura-
tion
spaces
associated
to
a
stable
log
curve
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0].
Roughly
speaking,
these
tripods
either
oc-
cur
in
the
original
stable
log
curve
or
arise
as
the
result
of
blowing
up
various
cusps
or
nodes
that
occur
in
the
one-dimensional
fibers
of
log
configuration
spaces
of
lower
dimension
[cf.
Figure
1
at
the
end
of
the
present
Introduction].
In
fact,
a
substantial
portion
of
§3
is
devoted
precisely
to
the
theory
of
classification
of
the
various
tripods
that
occur
in
the
one-dimensional
fibers
of
the
log
configuration
spaces
associated
to
a
stable
log
curve
[cf.
Lemmas
3.6,
3.8].
This
leads
natu-
rally
to
the
study
of
the
phenomenon
of
tripod
synchronization,
i.e.,
roughly
speaking,
the
phenomenon
that
an
outomorphism
[that
is
to
10
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
say,
an
outer
automorphism]
of
the
pro-Σ
fundamental
group
of
a
log
configuration
space
associated
to
a
stable
log
curve
typically
induces
the
same
outer
automorphism
on
the
various
[group-theoretic]
tripods
that
occur
in
subquotients
of
such
a
fundamental
group
[cf.
Theo-
rems
3.16,
3.17,
3.18].
The
phenomenon
of
tripod
synchronization,
in
turn,
leads
naturally
to
the
definition
of
the
tripod
homomorphism
[cf.
Definition
3.19],
which
may
be
thought
of
as
the
homomorphism
obtained
by
associating
to
an
[FC-admissible]
outer
automorphism
of
the
pro-Σ
fundamental
group
of
the
n-th
log
configuration
space
as-
sociated
to
a
stable
log
curve,
where
n
≥
3
is
a
positive
integer,
the
outer
automorphism
induced
on
a
[group-theoretic]
central
tripod,
i.e.,
roughly
speaking,
a
tripod
that
arises,
in
the
case
where
n
=
3
and
the
given
stable
log
curve
has
no
nodes,
by
blowing
up
the
intersection
of
the
three
diagonal
divisors
of
the
direct
product
of
three
copies
of
the
curve.
Theorem
C
(Synchronization
of
tripods
in
three
or
more
di-
mensions).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
n
a
positive
integer;
Σ
a
set
of
prime
numbers
which
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardinality
one;
k
an
algebraically
closed
field
of
characteristic
∈
Σ;
(Spec
k)
log
the
log
scheme
obtained
by
equipping
Spec
k
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
=
X
1
log
a
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
.
Write
G
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
determined
by
the
stable
log
curve
X
log
.
For
each
positive
integer
i,
write
X
i
log
for
the
i-th
log
configuration
space
of
the
stable
log
curve
X
log
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”];
Π
i
for
the
maximal
pro-Σ
quotient
of
the
kernel
of
the
natural
surjection
π
1
(X
i
log
)
π
1
((Spec
k)
log
).
Let
T
⊆
Π
m
be
a
{1,
·
·
·
,
m}-tripod
of
Π
n
[cf.
Definition
3.3,
(i)]
for
m
a
positive
integer
≤
n.
Suppose
that
n
≥
3.
Let
Π
tpd
⊆
Π
3
be
a
1-central
{1,
2,
3}-tripod
of
Π
n
[cf.
Definitions
3.3,
(i);
3.7,
(ii)].
Then
the
following
hold:
(i)
The
commensurator
and
centralizer
of
T
in
Π
m
satisfy
the
equality
C
Π
m
(T
)
=
T
×
Z
Π
m
(T
)
.
Thus,
if
an
outomorphism
α
of
Π
m
preserves
the
Π
m
-conjugacy
class
of
T
⊆
Π
m
,
then
one
obtains
a
“restriction”
α|
T
∈
Out(T
).
(ii)
Let
α
∈
Out
FC
(Π
n
)
be
an
FC-admissible
outomorphism
of
Π
n
.
Then
the
outomorphism
of
Π
3
induced
by
α
preserves
the
Π
3
-
conjugacy
class
of
Π
tpd
⊆
Π
3
.
In
particular,
by
(i),
we
obtain
COMBINATORIAL
ANABELIAN
TOPICS
II
11
a
natural
homomorphism
T
Π
tpd
:
Out
FC
(Π
n
)
−→
Out(Π
tpd
)
.
We
shall
refer
to
this
homomorphism
as
the
tripod
homo-
morphism
associated
to
Π
n
.
(iii)
Let
α
∈
Out
FC
(Π
n
)
be
an
FC-admissible
outomorphism
of
Π
n
such
that
the
outomorphism
α
m
of
Π
m
induced
by
α
preserves
the
Π
m
-conjugacy
class
of
T
⊆
Π
m
and
induces
[cf.
(i)]
the
identity
automorphism
of
the
set
of
T
-conjugacy
classes
of
cuspidal
inertia
subgroups
of
T
.
Then
there
exists
a
geometric
∼
[cf.
Definition
3.4,
(ii)]
outer
isomorphism
Π
tpd
→
T
with
respect
to
which
the
outomorphism
T
Π
tpd
(α)
∈
Out(Π
tpd
)
[cf.
(ii)]
is
compatible
with
the
outomorphism
α
m
|
T
∈
Out(T
)
[cf.
(i)].
(iv)
Suppose,
moreover,
that
either
n
≥
4
or
r
=
0.
Then
the
homomorphism
T
Π
tpd
of
(ii)
factors
through
Out
C
(Π
tpd
)
Δ+
⊆
Out(Π
tpd
)
[cf.
Definition
3.4,
(i)],
and,
moreover,
the
resulting
homomorphism
T
Π
tpd
:
Out
F
(Π
n
)
=
Out
FC
(Π
n
)
−→
Out
C
(Π
tpd
)
Δ+
[cf.
Theorem
A,
(ii)]
is
surjective.
Here,
we
remark
that
the
surjectivity
of
the
tripod
homomorphism
[cf.
Theorem
C,
(iv)]
is
obtained
[cf.
Corollary
4.15]
as
a
consequence
of
the
theory
of
glueability
of
combinatorial
cuspidalizations
developed
in
§4
[cf.
the
discussion
preceding
Theorem
F
below].
Also,
we
recall
that
the
codomain
of
this
surjective
tripod
homomorphism
Out
C
(Π
tpd
)
Δ+
may
be
identified
with
the
[pro-Σ]
Grothendieck-Teichmüller
group
GT
Σ
[cf.
the
discussion
of
[CmbCsp],
Remark
1.11.1].
Since
GT
Σ
may
be
thought
of
as
a
sort
of
abstract
combinatorial
approximation
of
the
absolute
Galois
group
G
Q
of
the
rational
number
field
Q,
it
is
thus
natural
to
think
of
the
surjective
tripod
homomorphism
Out
F
(Π
n
)
Out
C
(Π
tpd
)
Δ+
of
Theorem
C,
(iv),
as
a
sort
of
abstract
combinatorial
version
of
the
natural
surjective
outer
homomorphism
π
1
((M
g,[r]
)
Q
)
G
Q
induced
on
étale
fundamental
groups
by
the
structure
morphism
(M
g,[r]
)
Q
→
Spec
(Q)
of
the
moduli
stack
(M
g,[r]
)
Q
of
hyperbolic
curves
of
type
(g,
r)
[cf.
the
discussion
of
Remark
3.19.1].
In
particular,
the
kernel
of
the
tripod
homomorphism
—
which
we
denote
by
Out
F
(Π
n
)
geo
12
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
—
may
be
thought
of
as
a
sort
of
abstract
combinatorial
analogue
of
the
geometric
étale
fundamental
group
of
(M
g,[r]
)
Q
[i.e.,
the
kernel
of
the
natural
outer
homomorphism
π
1
((M
g,[r]
)
Q
)
G
Q
].
One
interesting
application
of
the
theory
of
tripod
synchronization
is
the
following.
Fix
a
pro-Σ
fundamental
group
of
a
hyperbolic
curve.
Recall
the
notion
of
a
nondegenerate
profinite
Dehn
multi-twist
[cf.
[CbTpI],
Definition
4.4;
[CbTpI],
Definition
5.8,
(ii)]
associated
to
a
structure
of
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
on
such
a
fun-
damental
group.
Here,
we
recall
that
such
a
structure
may
be
thought
of
as
a
sort
of
profinite
analogue
of
the
notion
of
a
decomposition
of
a
hyperbolic
topological
surface
into
hyperbolic
subsurfaces
[i.e.,
such
as
“pants”].
Then
the
following
result
asserts
that,
under
certain
techni-
cal
conditions,
any
such
nondegenerate
profinite
Dehn
multi-twist
that
commutes
with
another
nondegenerate
profinite
Dehn
multi-twist
as-
sociated
to
some
given
totally
degenerate
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
[cf.
[CbTpI],
Definition
2.3,
(iv)]
necessarily
arises
from
a
structure
of
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
that
is
“co-Dehn”
to,
i.e.,
arises
by
applying
a
deformation
to,
the
given
totally
degenerate
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
[cf.
Corollary
3.25].
This
sort
of
result
is
reminiscent
of
topological
results
concerning
subgroups
of
the
mapping
class
group
generated
by
pairs
of
positive
Dehn
multi-twists
[cf.
[Ishi],
[HT]].
Theorem
D
(Co-Dehn-ness
of
degeneration
structures
in
the
totally
degenerate
case).
In
the
notation
of
Theorem
C,
for
i
=
1,
2,
let
Y
i
log
be
a
stable
log
curve
over
(Spec
k)
log
;
H
i
the
“G”
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
Y
i
log
;
(H
i
,
S
i
,
φ
i
)
a
3-
cuspidalizable
degeneration
structure
on
G
[cf.
Definition
3.23,
(i),
(v)];
α
i
∈
Out(Π
G
)
a
nondegenerate
(H
i
,
S
i
,
φ
i
)-Dehn
multi-twist
of
G
[cf.
Definition
3.23,
(iv)].
Suppose
that
α
1
commutes
with
α
2
,
and
that
H
2
is
totally
degenerate
[cf.
[CbTpI],
Definition
2.3,
(iv)].
Suppose,
moreover,
that
one
of
the
following
conditions
is
satisfied:
(i)
r
=
0.
(ii)
α
1
and
α
2
are
positive
definite
[cf.
Definition
3.23,
(iv)].
Then
(H
1
,
S
1
,
φ
1
)
is
co-Dehn
to
(H
2
,
S
2
,
φ
2
)
[cf.
Definition
3.23,
(iii)],
or,
equivalently
[since
H
2
is
totally
degenerate],
(H
2
,
S
2
,
φ
2
)
(H
1
,
S
1
,
φ
1
)
[cf.
Definition
3.23,
(ii)].
Another
interesting
application
of
the
theory
of
tripod
synchroniza-
tion
is
to
the
computation,
in
terms
of
a
certain
scheme-theoretic
fundamental
group,
of
the
purely
combinatorial
commensurator
of
the
subgroup
of
profinite
Dehn
multi-twists
in
the
group
of
3-cuspidali-
zable,
FC-admissible,
“geometric”
outer
automorphisms
of
the
pro-Σ
COMBINATORIAL
ANABELIAN
TOPICS
II
13
fundamental
group
of
a
totally
degenerate
stable
log
curve
[cf.
Corol-
lary
3.27].
Here,
we
remark
that
the
scheme-theoretic
[or,
perhaps
more
precisely,
“log
algebraic
stack-theoretic”]
fundamental
group
that
ap-
pears
is,
roughly
speaking,
the
pro-Σ
geometric
fundamental
group
of
a
formal
neighborhood,
in
the
corresponding
logarithmic
moduli
stack,
of
the
point
determined
by
the
given
totally
degenerate
sta-
ble
log
curve.
In
particular,
this
computation
may
also
be
regarded
as
a
sort
of
purely
combinatorial
algorithm
for
constructing
this
scheme-theoretic
fundamental
group
[cf.
Remark
3.27.1].
Theorem
E
(Commensurator
of
profinite
Dehn
multi-twists
in
the
totally
degenerate
case).
In
the
notation
of
Theorem
C
[so
n
≥
3],
suppose
further
that
if
r
=
0,
then
n
≥
4.
Also,
we
assume
that
G
is
totally
degenerate
[cf.
[CbTpI],
Definition
2.3,
def
(iv)].
Write
s
:
Spec
k
→
(M
g,[r]
)
k
=
(M
g,[r]
)
Spec
k
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”]
for
the
underly-
ing
(1-)morphism
of
algebraic
stacks
of
the
classifying
(1-)morphism
log
log
def
(Spec
k)
log
→
(M
g,[r]
)
k
=
(M
g,[r]
)
Spec
k
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”]
of
the
stable
log
curve
X
log
log
for
the
log
scheme
obtained
by
equipping
N
s
def
over
(Spec
k)
log
;
N
=
s
Spec
k
with
the
log
structure
induced,
via
s,
by
the
log
structure
of
log
(M
g,[r]
)
k
;
N
s
log
for
the
log
stack
obtained
by
forming
the
[stack-theoretic]
log
by
the
natural
action
of
the
finite
k-
quotient
of
the
log
scheme
N
s
group
“s
×
(M
g,[r]
)
k
s”,
i.e.,
the
fiber
product
over
(M
g,[r]
)
k
of
two
copies
of
s;
N
s
for
the
underlying
stack
of
the
log
stack
N
s
log
;
I
N
s
⊆
π
1
(N
s
log
)
for
the
closed
subgroup
of
the
log
fundamental
group
π
1
(N
s
log
)
of
N
s
log
given
by
the
kernel
of
the
natural
surjection
π
1
(N
s
log
)
π
1
(N
s
)
[in-
duced
by
the
(1-)morphism
N
s
log
→
N
s
obtained
by
forgetting
the
log
(Σ)
structure];
π
1
(N
s
log
)
for
the
quotient
of
π
1
(N
s
log
)
by
the
kernel
of
the
Σ
natural
surjection
from
I
N
s
to
its
maximal
pro-Σ
quotient
I
N
.
Then
s
we
have
an
equality
N
Out
F
(Π
n
)
geo
(Dehn(G))
=
C
Out
F
(Π
n
)
geo
(Dehn(G))
and
a
natural
commutative
diagram
of
profinite
groups
1
−−−→
Σ
I
N
⏐
s
⏐
−−−→
(Σ)
π
1
(N
s
log
)
⏐
⏐
−−−→
π
1
(N
s
)
−−−→
1
⏐
⏐
1
−−−→
Dehn(G)
−−−→
C
Out
F
(Π
n
)
geo
(Dehn(G))
−−−→
Aut(G)
−−−→
1
[cf.
Definition
3.1,
(ii),
concerning
the
notation
“G”]
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
isomor-
phisms.
Moreover,
Dehn(G)
is
open
in
C
Out
F
(Π
n
)
geo
(Dehn(G)).
14
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
In
§4,
we
show,
under
suitable
technical
conditions,
that
an
auto-
morphism
of
the
pro-Σ
fundamental
group
of
the
log
configuration
space
associated
to
a
stable
log
curve
necessarily
preserves
the
graph-
theoretic
structure
of
the
various
one-dimensional
fibers
of
such
a
log
configuration
space
[cf.
Theorem
4.7].
This
allows
us
to
verify
the
glueability
of
combinatorial
cuspidalizations,
i.e.,
roughly
speak-
ing,
that,
for
n
≥
2
a
positive
integer,
the
datum
of
an
n-cuspidalizable
outer
automorphism
of
the
pro-Σ
fundamental
group
of
a
stable
log
curve
is
equivalent,
up
to
possible
composition
with
a
profinite
Dehn
multi-twist,
to
the
datum
of
a
collection
of
n-cuspidalizable
automor-
phisms
of
the
pro-Σ
fundamental
groups
of
the
various
irreducible
com-
ponents
of
the
given
stable
log
curve
that
satisfy
a
certain
gluing
condi-
tion
involving
the
induced
outer
actions
on
tripods
[cf.
Theorem
4.14].
Theorem
F
(Glueability
of
combinatorial
cuspidalizations).
In
the
notation
of
Theorem
C,
write
Out
FC
(Π
n
)
brch
⊆
Out
FC
(Π
n
)
for
the
closed
subgroup
of
Out
FC
(Π
n
)
consisting
of
FC-admissible
out-
omorphisms
α
of
Π
n
such
that
the
outomorphism
of
Π
1
determined
by
α
induces
the
identity
automorphism
of
Vert(G),
Node(G),
and,
more-
over,
fixes
each
of
the
branches
of
every
node
of
G
[cf.
Definition
4.6,
(i)];
Out
FC
((Π
v
)
n
)
Glu(Π
n
)
⊆
v∈Vert(G)
for
the
closed
subgroup
of
v∈Vert(G)
Out
FC
((Π
v
)
n
)
consisting
of
“glue-
able”
collections
of
outomorphisms
of
the
groups
“(Π
v
)
n
”
[cf.
Defini-
tion
4.9,
(iii)].
Then
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Dehn(G)
−→
Out
FC
(Π
n
)
brch
−→
Glu(Π
n
)
−→
1
.
This
glueability
result
may,
alternatively,
be
thought
of
as
a
re-
sult
that
asserts
the
localizability
[i.e.,
relative
to
localization
on
the
dual
semi-graph
of
the
given
stable
log
curve]
of
the
notion
of
n-
cuspidalizability.
In
this
context,
it
is
of
interest
to
observe
that
this
glueability
result
may
be
regarded
as
a
natural
generalization,
to
the
case
of
n-cuspidalizability
for
n
≥
2,
of
the
glueability
result
obtained
in
[CbTpI],
Theorem
B,
(iii),
in
the
“1-cuspidalizable”
case,
which
is
derived
as
a
consequence
of
the
theory
of
localizability
[i.e.,
relative
to
localization
on
the
dual
semi-graph
of
the
given
stable
log
curve]
and
synchronization
of
cyclotomes
developed
in
[CbTpI],
§3,
§4.
From
this
point
of
view,
it
is
also
of
interest
to
observe
that
the
sufficiency
portion
of
[the
equivalence
that
constitutes]
this
glueability
result
[i.e.,
Theorem
F]
may
be
thought
of
as
a
sort
of
“converse”
to
the
theory
of
tripod
synchronizations
developed
in
§3
[i.e.,
of
which
the
necessity
COMBINATORIAL
ANABELIAN
TOPICS
II
15
portion
of
this
glueability
result
is,
in
essence,
a
formal
consequence
—
cf.
the
proof
of
Lemma
4.10,
(ii)].
Indeed,
the
bulk
of
the
proof
given
in
§4
of
Theorem
4.14
is
devoted
to
the
sufficiency
portion
of
this
result,
which
is
verified
by
means
of
a
detailed
combinatorial
analysis
[cf.
the
proof
of
[CbTpI],
Proposition
4.10,
(ii)]
of
the
noncyclically
primi-
tive
and
cyclically
primitive
cases
[cf.
Lemmas
4.12,
4.13;
Figures
2,
3,
4].
Finally,
we
apply
this
glueability
result
to
derive
a
cuspidalization
theorem
—
i.e.,
in
the
spirit
of
and
generalizing
the
corresponding
results
of
[AbsCsp],
Theorem
3.1;
[Hsh],
Theorem
0.1;
[Wkb],
Theorem
C
[cf.
Remark
4.16.1]
—
for
geometrically
pro-l
fundamental
groups
of
stable
log
curves
over
finite
fields
[cf.
Corollary
4.16].
That
is
to
say,
in
the
case
of
stable
log
curves
over
finite
fields,
the
condition
of
compatibility
with
the
Galois
action
is
sufficient
to
imply
the
n-cuspidalizability
of
arbi-
trary
isomorphisms
between
the
geometric
pro-l
fun-
damental
groups,
for
n
≥
1.
In
this
context,
it
is
of
interest
to
recall
that
strong
anabelian
results
[i.e.,
in
the
style
of
the
“Grothendieck
Conjecture”]
for
such
geomet-
rically
pro-l
fundamental
groups
of
stable
log
curves
over
finite
fields
are
not
known
in
general,
at
the
time
of
writing.
On
the
other
hand,
we
observe
that
in
the
case
of
totally
degenerate
stable
log
curves
over
finite
fields,
such
“strong
anabelian
results”
may
be
obtained
un-
der
certain
technical
conditions
[cf.
Corollary
4.17;
Remarks
4.17.1,
4.17.2].
Notations
and
Conventions
Sets:
If
S
is
a
set,
then
we
shall
denote
by
#S
the
cardinality
of
S.
Groups:
We
shall
refer
to
an
element
of
a
group
as
trivial
(respectively,
nontrivial)
if
it
is
(respectively,
is
not)
equal
to
the
identity
element
of
the
group.
We
shall
refer
to
a
nonempty
subset
of
a
group
as
trivial
(respectively,
nontrivial)
if
it
is
(respectively,
is
not)
equal
to
the
set
whose
unique
element
is
the
identity
element
of
the
group.
Topological
groups:
Let
G
be
a
topological
group
and
J,
H
⊆
G
closed
subgroups.
Then
we
shall
write
def
Z
J
(H)
=
{
j
∈
J
|
jh
=
hj
for
any
h
∈
H
}
=
Z
G
(H)
∩
J
for
the
centralizer
of
H
in
J,
def
Z(G)
=
Z
G
(G)
for
the
center
of
G,
and
def
Z
J
loc
(H)
=
lim
Z
J
(U
)
⊆
J
−→
16
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
—
where
the
inductive
limit
is
over
all
open
subgroups
U
⊆
H
of
H
—
def
for
the
“local
centralizer”
of
H
in
J.
We
shall
write
Z
loc
(G)
=
Z
G
loc
(G)
for
the
“local
center”
of
G.
Thus,
a
profinite
group
G
is
slim
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
if
and
only
if
Z
loc
(G)
=
{1}.
Rings:
If
R
is
a
commutative
ring
with
unity,
then
we
shall
write
R
∗
for
the
multiplicative
group
of
invertible
elements
of
R.
Curves:
Let
g,
r
1
,
r
2
be
nonnegative
integers
such
that
2g
−
2
+
r
1
+
r
2
>
0.
Then
we
shall
write
M
g,[r
1
]+r
2
for
the
moduli
stack
of
pointed
stable
curves
of
type
(g,
r
1
+
r
2
),
where
the
first
r
1
marked
points
are
regarded
as
unordered,
but
the
last
r
2
marked
points
are
regarded
as
ordered,
over
Z;
M
g,[r
1
]+r
2
⊆
M
g,[r
1
]+r
2
for
the
open
sub-
log
stack
of
M
g,[r
1
]+r
2
that
parametrizes
smooth
curves;
M
g,[r
1
]+r
2
for
the
log
stack
obtained
by
equipping
M
g,[r
1
]+r
2
with
the
log
structure
as-
sociated
to
the
divisor
with
normal
crossings
M
g,[r
1
]+r
2
\
M
g,[r
1
]+r
2
⊆
M
g,[r
1
]+r
2
;
C
g,[r
1
]+r
2
→
M
g,[r
1
]+r
2
for
the
tautological
stable
curve
over
M
g,[r
1
]+r
2
;
D
g,[r
1
]+r
2
⊆
C
g,[r
1
]+r
2
for
the
corresponding
tautological
di-
visor
of
cusps
of
C
g,[r
1
]+r
2
→
M
g,[r
1
]+r
2
.
Then
the
divisor
given
by
the
union
of
D
g,[r
1
]+r
2
with
the
inverse
image
in
C
g,[r
1
]+r
2
of
the
divi-
sor
M
g,[r
1
]+r
2
\
M
g,[r
1
]+r
2
⊆
M
g,[r
1
]+r
2
determines
a
log
structure
on
log
C
g,[r
1
]+r
2
;
write
C
g,[r
1
]+r
2
for
the
resulting
log
stack.
Thus,
we
obtain
log
log
a
(1-)morphism
of
log
stacks
C
g,[r
1
]+r
2
→
M
g,[r
1
]+r
2
.
We
shall
write
log
C
g,[r
1
]+r
2
⊆
C
g,[r
1
]+r
2
for
the
interior
of
C
g,[r
1
]+r
2
[cf.
the
discussion
entitled
“Log
schemes”
in
[CbTpI],
§0].
In
particular,
we
obtain
a
(1-)morphism
of
stacks
C
g,[r
1
]+r
2
→
M
g,[r
1
]+r
2
.
Moreover,
for
a
nonneg-
def
ative
integer
r
such
that
2g−2+r
>
0,
we
shall
write
M
g,[r]
=
M
g,[r]+0
;
log
def
def
log
def
def
M
g,[r]
=
M
g,[r]+0
;
M
g,[r]
=
M
g,[r]+0
;
C
g,[r]
=
C
g,[r]+0
;
D
g,[r]
=
D
g,[r]+0
;
log
def
log
def
C
g,[r]
=
C
g,[r]+0
;
C
g,[r]
=
C
g,[r]+0
.
In
particular,
the
stack
M
g,[r]
may
be
regarded
as
a
moduli
stack
of
hyperbolic
curves
of
type
(g,
r)
over
Z.
If
S
is
a
scheme,
then
we
shall
denote
by
means
of
a
subscript
S
the
result
of
base-changing
via
the
structure
morphism
S
→
Spec
Z
the
various
log
stacks
of
the
above
discussion.
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
n
a
positive
integer;
X
log
a
stable
log
curve
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0]
of
type
(g,
r)
over
a
log
scheme
S
log
.
Then
we
shall
refer
to
the
log
scheme
obtained
by
pulling
back
the
(1-)morphism
log
log
M
g,[r]+n
→
M
g,[r]
given
by
forgetting
the
last
n
[ordered]
points
via
the
classifying
(1-)morphism
S
log
→
M
g,[r]
of
X
log
as
the
n-th
log
conguration
space
of
X
log
.
COMBINATORIAL
ANABELIAN
TOPICS
II
17
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
tripod
.
.
.
.
.
.
.
.
.
tripod
tripod
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
tripod
tripod
tripod
Figure
1
:
tripods
in
the
various
fibers
of
a
configuration
space
18
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
1.
Combinatorial
anabelian
geometry
in
the
absence
of
group-theoretic
cuspidality
In
the
present
§1,
we
discuss
various
combinatorial
versions
of
the
Grothendieck
Conjecture
for
outer
representations
of
NN-
and
IPSC-
type
[cf.
Theorem
1.9
below].
These
Grothendieck
Conjecture-type
results
may
be
regarded
as
generalizations
of
[NodNon],
Corollary
4.2;
[NodNon],
Remark
4.2.1,
that
may
be
applied
to
isomorphisms
that
are
not
necessarily
group-theoretically
cuspidal.
For
instance,
we
prove
[cf.
Theorem
1.9,
(ii),
below]
that
any
isomorphism
between
outer
representations
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)]
is
nec-
essarily
group-theoretically
verticial,
i.e.,
roughly
speaking,
preserves
the
verticial
subgroups.
A
basic
reference
for
the
theory
of
semi-graphs
of
anabelioids
of
PSC-
type
is
[CmbGC].
We
shall
use
the
terms
“semi-graph
of
anabelioids
of
PSC-type”,
“PSC-fundamental
group
of
a
semi-graph
of
anabelioids
of
PSC-type”,
“finite
étale
covering
of
semi-graphs
of
anabelioids
of
PSC-
type”,
“vertex”,
“edge”,
“node”,
“cusp”,
“verticial
subgroup”,
“edge-like
subgroup”,
“nodal
subgroup”,
“cuspidal
subgroup”,
and
“sturdy”
as
they
are
defined
in
[CmbGC],
Definition
1.1
[cf.
also
Remark
1.1.2
below].
Also,
we
shall
apply
the
various
notational
conventions
established
in
[NodNon],
Definition
1.1,
and
refer
to
the
“PSC-fundamental
group
of
a
semi-graph
of
anabelioids
of
PSC-type”
simply
as
the
“fundamental
group”
[of
the
semi-graph
of
anabelioids
of
PSC-type].
That
is
to
say,
we
shall
refer
to
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
[as
a
semi-
graph
of
anabelioids!]
as
the
“fundamental
group
of
the
semi-graph
of
anabelioids
of
PSC-type”.
In
the
present
§1,
let
Σ
be
a
nonempty
set
of
prime
numbers
and
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type.
Write
G
for
the
under-
lying
semi-graph
of
G,
Π
G
for
the
[pro-Σ]
fundamental
group
of
G,
and
G
→
G
for
the
universal
covering
of
G
corresponding
to
Π
G
.
Then
since
the
fundamental
group
Π
G
of
G
is
topologically
finitely
generated,
the
profinite
topology
of
Π
G
induces
[profinite]
topologies
on
Aut(Π
G
)
and
Out(Π
G
)
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
If,
moreover,
we
write
Aut(G)
for
the
automorphism
group
of
G,
then,
by
the
discussion
preceding
[CmbGC],
Lemma
2.1,
the
natural
homomorphism
Aut(G)
−→
Out(Π
G
)
is
an
injection
with
closed
image.
[Here,
we
recall
that
an
automor-
phism
of
a
semi-graph
of
anabelioids
consists
of
an
automorphism
of
the
underlying
semi-graph,
together
with
a
compatible
system
of
iso-
morphisms
between
the
various
anabelioids
at
each
of
the
vertices
and
COMBINATORIAL
ANABELIAN
TOPICS
II
19
edges
of
the
underlying
semi-graph
which
are
compatible
with
the
var-
ious
morphisms
of
anabelioids
associated
to
the
branches
of
the
under-
lying
semi-graph
—
cf.
[SemiAn],
Definition
2.1;
[SemiAn],
Remark
2.4.2.]
Thus,
by
equipping
Aut(G)
with
the
topology
induced
via
this
homomorphism
by
the
topology
of
Out(Π
G
),
we
may
regard
Aut(G)
as
being
equipped
with
the
structure
of
a
profinite
group.
Definition
1.1.
We
shall
say
that
an
element
γ
∈
Π
G
of
Π
G
is
verticial
(respectively,
edge-like;
nodal;
cuspidal)
if
γ
is
contained
in
a
verticial
(respectively,
an
edge-like;
a
nodal;
a
cuspidal)
subgroup
of
Π
G
.
Remark
1.1.1.
Let
γ
∈
Π
G
be
a
nontrivial
[cf.
the
discussion
entitled
“Groups”
in
“Notations
and
Conventions”]
element
of
Π
G
.
If
γ
∈
Π
G
is
edge-like
[cf.
Definition
1.1],
then
it
follows
from
[NodNon],
Lemma
1.5,
such
that
γ
∈
Π
e
.
If
γ
∈
Π
G
that
there
exists
a
unique
edge
e
∈
Edge(
G)
is
verticial,
but
not
nodal
[cf.
Definition
1.1],
then
it
follows
from
[NodNon],
Lemma
1.9,
(i),
that
there
exists
a
unique
vertex
v
∈
Vert(
G)
such
that
γ
∈
Π
v
.
Remark
1.1.2.
Here,
we
take
the
opportunity
to
correct
an
unfortu-
nate
misprint
in
[CmbGC].
In
the
final
sentence
of
[CmbGC],
Definition
1.1,
(ii),
the
phrase
“rank
≥
2”
should
read
“rank
>
2”.
In
particular,
we
shall
say
that
G
is
sturdy
if
the
abelianization
of
the
image,
in
the
quotient
Π
G
Π
unr
of
Π
G
by
the
normal
closed
subgroup
normally
G
topologically
generated
by
the
edge-like
subgroups,
of
every
verticial
subgroup
of
Π
G
is
free
of
rank
>
2
over
Z
Σ
.
Here,
we
note
in
passing
that
G
is
sturdy
if
and
only
if
every
vertex
of
G
is
of
genus
≥
2
[cf.
[CbTpI],
Definition
2.3,
(iii)].
Lemma
1.2
(Existence
of
a
certain
connected
finite
étale
cov-
ering).
Let
n
be
a
positive
integer
which
is
a
product
[possibly
with
v
∈
Vert(
G).
Write
multiplicities!]
of
primes
∈
Σ;
e
1
,
e
2
∈
Edge(
G);
def
def
def
e
1
=
e
1
(G),
e
2
=
e
2
(G),
and
v
=
v
(G).
Suppose
that
the
following
conditions
are
satisfied:
(i)
G
is
untangled
[cf.
[NodNon],
Definition
1.2].
(ii)
If
e
1
is
a
node,
then
the
following
condition
holds:
Let
w,
w
∈
V(e
1
)
be
the
two
distinct
elements
of
V(e
1
)
[cf.
(i)].
Then
#(N
(w)
∩
N
(w
))
≥
3.
(iii)
If
e
1
is
a
cusp,
then
the
following
condition
holds:
Let
w
∈
V(e
1
)
be
the
unique
element
of
V(e
1
).
Then
#C(w)
≥
3.
20
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(iv)
e
1
=
e
2
.
(v)
v
∈
V(e
1
).
Then
there
exists
a
finite
étale
Galois
subcovering
G
→
G
of
G
→
G
such
that
n
divides
[Π
e
1
:
Π
e
1
∩
Π
G
],
and,
moreover,
Π
e
2
,
Π
v
⊆
Π
G
.
Proof.
Suppose
that
e
1
is
a
node
(respectively,
cusp).
Write
H
for
the
[uniquely
determined]
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
of
G
whose
set
of
vertices
is
=
V(e
1
)
=
{w,
w
}
[cf.
condition
(ii)]
(respectively,
=
{w}
[cf.
condition
(iii)]).
Now
it
follows
from
condition
(ii)
(respectively,
(iii))
that
there
exists
an
e
3
∈
Node(G|
H
)
=
N
(w)
∩
N
(w
)
(respectively,
∈
Cusp(G|
H
)
∩
Cusp(G)
=
C(w))
[cf.
[CbTpI],
Definition
2.2,
(ii)]
such
that
e
3
=
e
2
.
Moreover,
again
by
applying
condition
(ii)
(respectively,
(iii)),
together
with
the
well-known
structure
of
the
abelianization
of
the
fundamental
group
of
a
smooth
curve
over
an
algebraically
closed
field
of
characteristic
∈
Σ,
we
conclude
that
there
exists
a
finite
étale
Galois
covering
G
H
→
G|
H
that
arises
from
a
normal
open
subgroup
of
Π
G|
H
and
which
is
unramified
at
every
element
of
Edge(G|
H
)
\
{e
1
,
e
3
}
and
totally
ramified
at
e
1
,
e
3
with
ramification
indices
divisible
by
n.
Now
since
G
H
→
G|
H
is
unramified
at
every
element
of
Cusp(G|
H
)
∩
Node(G),
one
may
extend
this
covering
to
a
finite
étale
Galois
subcovering
G
→
G
of
G
→
G
which
restricts
to
the
trivial
covering
over
every
vertex
u
of
G
such
that
u
=
w,
w
(respectively,
u
=
w).
Moreover,
it
follows
immediately
from
the
construction
of
G
→
G
that
n
divides
[Π
e
1
:
Π
e
1
∩
Π
G
],
and
Π
e
2
,
Π
v
⊆
Π
G
.
This
completes
the
proof
of
Lemma
1.2.
Lemma
1.3
(Product
of
edge-like
elements).
Let
γ
1
,
γ
2
∈
Π
G
be
two
nontrivial
edge-like
elements
of
Π
G
[cf.
Definition
1.1].
Write
for
the
unique
elements
of
Edge(
G)
such
that
γ
1
∈
e
1
,
e
2
∈
Edge(
G)
Π
e
1
,
γ
2
∈
Π
e
2
[cf.
Remark
1.1.1].
Suppose
that
the
following
conditions
are
satisfied:
(i)
For
every
positive
integer
n,
it
holds
that
γ
1
n
γ
2
n
is
verticial.
(ii)
e
1
=
e
2
.
Then
there
exists
a
[necessarily
unique
—
cf.
[NodNon],
Remark
1.8.1,
such
that
{
v
);
in
particular,
it
holds
that
(iii)]
v
∈
Vert(
G)
e
1
,
e
2
}
⊆
E(
γ
1
γ
2
∈
Π
v
.
Proof.
Since
e
1
=
e
2
[cf.
condition
(ii)],
one
verifies
easily
that
there
exists
a
finite
étale
Galois
subcovering
H
→
G
of
G
→
G
that
satisfies
the
following
conditions:
(1)
e
1
(H)
=
e
2
(H).
COMBINATORIAL
ANABELIAN
TOPICS
II
21
(2)
H
is
untangled
[cf.
[NodNon],
Definition
1.2;
[NodNon],
Re-
mark
1.2.1,
(i),
(ii)].
then
the
following
holds:
Let
(3)
For
i
∈
{1,
2},
if
e
i
∈
Node(
G),
e
i
(H))
be
the
two
distinct
elements
of
V(
e
i
(H))
[cf.
w,
w
∈
V(
(ii)].
Then
#(N
(w)
∩
N
(w
))
≥
3.
then
the
following
holds:
(4)
For
i
∈
{1,
2},
if
e
i
∈
Cusp(
G),
Let
w
∈
V(
e
i
(H))
be
the
unique
element
of
V(
e
i
(H)).
Then
#C(w)
≥
3.
Now
it
is
immediate
that
there
exists
a
positive
integer
m
such
that
be
such
that
γ
1
m
γ
2
m
∈
γ
1
m
∈
Π
e
1
∩
Π
H
,
γ
2
m
∈
Π
e
2
∩
Π
H
.
Let
v
∈
Vert(
G)
Π
v
[cf.
condition
(i)].
Suppose
that
v
(H)
∈
V(
e
1
(H)).
Then
it
follows
from
Lemma
1.2
that
there
exists
a
finite
étale
Galois
subcovering
H
→
H
of
G
→
H
such
that
γ
1
m
∈
Π
H
,
and,
moreover,
Π
e
2
∩Π
H
,
Π
v
∩Π
H
⊆
Π
H
.
But
this
implies
that
γ
2
m
,
γ
1
m
γ
2
m
∈
Π
H
,
hence
that
γ
1
m
∈
Π
H
,
a
contradiction.
In
particular,
it
holds
that
v
(H)
∈
V(
e
1
(H));
a
similar
argument
implies
e
1
(H))
∩
V(
e
2
(H))
=
∅.
Thus,
by
that
v
(H)
∈
V(
e
2
(H)),
hence
that
V(
applying
this
argument
to
a
suitable
system
of
connected
finite
étale
coverings
of
H,
we
conclude
that
V(
e
1
)∩V(
e
2
)
=
∅,
i.e.,
that
there
exists
such
that
{
a
v
∈
Vert(
G)
e
1
,
e
2
}
⊆
E(
v
).
Then
since
Π
e
1
,
Π
e
2
⊆
Π
v
,
it
follows
immediately
that
γ
1
γ
2
∈
Π
v
.
This
completes
the
proof
of
Lemma
1.3.
Proposition
1.4
(Group-theoretic
characterization
of
closed
subgroups
of
edge-like
subgroups).
Let
H
⊆
Π
G
be
a
closed
sub-
group
of
Π
G
.
Then
the
following
conditions
are
equivalent:
(i)
H
is
contained
in
an
edge-like
subgroup.
(ii)
An
open
subgroup
of
H
is
contained
in
an
edge-like
sub-
group.
(iii)
Every
element
of
H
is
edge-like
[cf.
Definition
1.1].
(iv)
There
exists
a
connected
finite
étale
subcovering
G
†
→
G
of
G
→
G
such
that
for
any
connected
finite
étale
subcovering
G
→
G
of
G
→
G
that
factors
through
G
†
→
G,
the
image
of
the
composite
ab/edge
H
∩
Π
G
→
Π
G
Π
G
ab/edge
—
where
we
write
Π
G
for
the
torsion-free
[cf.
[CmbGC],
Remark
1.1.4]
quotient
of
the
abelianization
Π
ab
G
by
the
closed
subgroup
topologically
generated
by
the
images
in
Π
ab
G
of
the
edge-like
subgroups
of
Π
G
—
is
trivial.
22
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Proof.
The
implications
(i)
⇒
(ii)
⇒
(iv)
are
immediate.
The
equiv-
alence
(iii)
⇔
(iv)
follows
immediately
from
[NodNon],
Lemma
1.6.
Thus,
to
complete
the
verification
of
Proposition
1.4,
it
suffices
to
ver-
ify
the
implication
(iii)
⇒
(i).
To
this
end,
suppose
that
condition
(iii)
holds.
First,
we
observe
that,
to
verify
the
implication
(iii)
⇒
(i),
it
suffices
to
verify
the
following
assertion:
Claim
1.4.A:
Let
γ
1
,
γ
2
∈
H
be
nontrivial
elements.
for
the
unique
elements
of
Write
e
1
,
e
2
∈
Edge(
G)
such
that
γ
1
∈
Π
e
1
,
γ
2
∈
Π
e
2
[cf.
Remark
1.1.1].
Edge(
G)
Then
e
1
=
e
2
.
To
verify
Claim
1.4.A,
let
us
observe
that
it
follows
from
condition
(iii)
that,
for
every
positive
integer
n,
it
holds
that
γ
1
n
γ
2
n
is
edge-like,
hence
verticial.
Thus,
it
follows
immediately
from
Lemma
1.3
that
there
such
that
{
v
);
in
particular,
exists
an
element
v
∈
Vert(
G)
e
1
,
e
2
}
⊆
E(
it
holds
that
γ
1
,
γ
2
∈
Π
v
.
Thus,
to
complete
the
verification
of
Claim
1.4.A,
we
may
assume
without
loss
of
generality
—
by
replacing
Π
G
,
H
by
Π
v
,
Π
v
∩
H,
respectively
—
that
Node(G)
=
∅
[so
e
1
,
e
2
∈
Cusp(
G)].
Moreover,
we
may
assume
without
loss
of
generality
—
by
replacing
Π
G
(respectively,
γ
1
,
γ
2
)
by
a
suitable
open
subgroup
of
Π
G
(respectively,
suitable
powers
of
γ
1
,
γ
2
)
—
that
#Cusp(G)
≥
4.
Thus,
it
follows
immediately
from
the
well-known
structure
of
the
abelianization
of
the
fundamental
group
of
a
smooth
curve
over
an
algebraically
closed
field
of
characteristic
∈
Σ
that
the
direct
product
of
any
3
cuspidal
inertia
subgroups
of
Π
G
associated
to
distinct
cusps
of
G
maps
injectively
to
the
abelianization
Π
ab
G
of
Π
G
.
In
particular,
since
γ
1
γ
2
is
edge-like,
hence
cuspidal,
we
conclude,
by
considering
the
cuspidal
inertia
subgroups
that
contain
γ
1
,
γ
2
,
and
γ
1
γ
2
,
that
e
1
=
e
2
.
This
completes
the
proof
of
Claim
1.4.A,
hence
also
of
the
implication
(iii)
⇒
(i).
This
completes
the
proof
of
Proposition
1.4.
Proposition
1.5
(Group-theoretic
characterization
of
closed
subgroups
of
verticial
subgroups).
Let
H
⊆
Π
G
be
a
closed
sub-
group
of
Π
G
.
Then
the
following
conditions
are
equivalent:
(i)
H
is
contained
in
a
verticial
subgroup.
(ii)
An
open
subgroup
of
H
is
contained
in
a
verticial
subgroup.
(iii)
Every
element
of
H
is
verticial
[cf.
Definition
1.1].
(iv)
There
exists
a
connected
finite
étale
subcovering
G
†
→
G
of
G
→
G
such
that
for
any
connected
finite
étale
subcovering
G
→
G
of
G
→
G
that
factors
through
G
†
→
G,
the
image
of
the
composite
H
∩
Π
G
→
Π
G
Π
ab-comb
G
COMBINATORIAL
ANABELIAN
TOPICS
II
23
—
where
we
write
Π
ab-comb
for
the
torsion-free
[cf.
[CmbGC],
G
Remark
1.1.4]
quotient
of
the
abelianization
Π
ab
G
by
the
closed
subgroup
topologically
generated
by
the
images
in
Π
ab
G
of
the
verticial
subgroups
of
Π
G
—
is
trivial.
Proof.
The
implications
(i)
⇒
(ii)
⇒
(iv)
are
immediate.
Next,
we
verify
the
implication
(iv)
⇒
(iii).
Suppose
that
condition
(iv)
holds.
Let
γ
∈
H.
Then
to
verify
that
γ
is
verticial,
we
may
assume
with-
out
loss
of
generality
—
by
replacing
H
by
the
procyclic
subgroup
of
H
topologically
generated
by
γ
—
that
H
is
procyclic.
Now
the
implication
(iv)
⇒
(iii)
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
implication
(ii)
⇒
(i)
of
[NodNon],
Lemma
1.6,
in
the
edge-like
case.
Here,
we
note
that
unlike
the
edge-like
case,
there
is
a
slight
complication
arising
from
the
fact
is
not
[cf.
[NodNon],
Lemma
1.9,
(i)]
that
an
element
v
∈
Vert(
G)
necessarily
uniquely
determined
by
the
condition
that
H
⊆
Π
v
,
i.e.,
such
that
e
)
for
some
e
∈
Node(
G)
there
may
exist
distinct
v
1
,
v
2
∈
V(
H
⊆
Π
e
=
Π
v
1
∩
Π
v
2
.
On
the
other
hand,
this
phenomenon
is,
in
fact,
irrelevant
to
the
argument
in
question,
since
Π
G
does
not
contain
any
elements
that
fix,
but
permute
the
branches
of,
e
.
This
completes
the
proof
of
the
implication
(iv)
⇒
(iii).
Finally,
we
verify
the
implication
(iii)
⇒
(i).
Suppose
that
condition
(iii)
holds.
Now
if
every
element
of
H
is
edge-like,
then
the
implication
(iii)
⇒
(i)
follows
from
the
implication
(iii)
⇒
(i)
of
Proposition
1.4,
together
with
the
fact
that
every
edge-like
subgroup
is
contained
in
a
verticial
subgroup.
Thus,
to
verify
the
implication
(iii)
⇒
(i),
we
may
assume
without
loss
of
generality
that
there
exists
an
element
γ
1
∈
H
for
the
unique
element
of
H
that
is
not
edge-like.
Write
v
1
∈
Vert(
G)
such
that
γ
1
∈
Π
v
1
[cf.
Remark
1.1.1].
of
Vert(
G)
Now
we
claim
the
following
assertion:
Claim
1.5.A:
H
⊆
Π
v
1
.
Indeed,
let
γ
2
∈
H
be
a
nontrivial
element
of
H.
If
γ
2
=
γ
1
,
then
γ
2
∈
Π
v
1
.
Thus,
we
may
assume
without
loss
of
generality
that
γ
1
=
γ
2
.
def
Write
γ
=
γ
1
γ
2
−1
.
for
the
Next,
suppose
that
γ
2
is
not
edge-like.
Write
v
2
∈
Vert(
G)
such
that
γ
2
∈
Π
v
2
[cf.
Remark
1.1.1].
Let
unique
element
of
Vert(
G)
H
→
G
be
a
connected
finite
étale
subcovering
of
G
→
G.
Then
since
neither
γ
1
nor
γ
2
is
edge-like,
one
verifies
easily
—
by
applying
the
implication
(iv)
⇒
(i)
of
Proposition
1.4
to
the
closed
subgroups
of
Π
G
topologically
generated
by
γ
1
,
γ
2
,
respectively
—
that
there
exist
a
connected
finite
étale
subcovering
H
→
H
of
G
→
H
and
a
positive
integer
n
such
that
γ
1
n
,
γ
2
n
∈
Π
H
⊆
Π
H
,
and,
moreover,
the
images
ab/edge
[cf.
the
of
γ
1
n
,
γ
2
n
∈
Π
H
via
the
natural
surjection
Π
H
Π
H
notation
of
Proposition
1.4,
(iv)]
are
nontrivial.
Thus,
it
follows
from
24
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
the
existence
of
the
natural
split
injection
ab/edge
Π
v
ab/edge
−→
Π
H
v∈Vert(H
)
of
[NodNon],
Lemma
1.4,
together
with
the
fact
that
γ
1
n
γ
2
n
∈
Π
H
is
verticial
[cf.
condition
(iii)],
that
v
1
(H
)
=
v
2
(H
),
hence
that
v
1
(H)
=
v
2
(H).
Therefore,
by
allowing
the
subcovering
H
→
G
of
G
→
G
to
vary,
we
conclude
that
v
1
=
v
2
;
in
particular,
it
holds
that
γ
2
∈
Π
v
1
.
Next,
suppose
that
γ
2
is
edge-like,
but
that
γ
is
not
edge-like.
Then,
by
applying
the
argument
of
the
preceding
paragraph
concerning
γ
2
to
γ,
we
conclude
that
γ,
hence
also
γ
2
,
is
contained
in
Π
v
1
.
Next,
suppose
that
both
γ
2
and
γ
are
edge-like.
Write
e
2
,
e
∈
for
the
unique
elements
of
Edge(
G)
such
that
γ
2
∈
Π
e
2
,
γ
∈
Π
e
Edge(
G)
[cf.
Remark
1.1.1].
Then
since
γ
1
is
not
edge-like,
it
follows
immedi-
ately
that
e
2
=
e
.
Moreover,
it
follows
from
condition
(iii)
that
for
any
positive
integer
n,
the
element
γ
2
n
γ
n
is
verticial.
Thus,
it
follows
immediately
from
Lemma
1.3
that
there
exists
a
unique
v
∈
Vert(
G)
such
that
{
e
2
,
e
}
⊆
E(
v
),
γ
1
=
γγ
2
∈
Π
v
.
On
the
other
hand,
since
is
uniquely
determined
by
the
condition
that
γ
1
∈
Π
v
1
,
we
v
1
∈
Vert(
G)
thus
conclude
that
v
1
=
v
,
hence
that
γ
2
∈
Π
e
2
⊆
Π
v
1
,
as
desired.
This
completes
the
proof
of
Claim
1.5.A
and
hence
also
of
the
implication
(iii)
⇒
(i).
Theorem
1.6
(Section
conjecture-type
result
for
outer
rep-
resentations
of
SNN-,
IPSC-type).
Let
Σ
be
a
nonempty
set
of
prime
numbers,
G
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type,
and
I
→
Aut(G)
an
outer
representation
of
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)].
Write
Π
G
for
the
[pro-Σ]
fundamental
group
of
G
def
out
and
Π
I
=
Π
G
I
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0];
thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
G
−→
Π
I
−→
I
−→
1
.
Write
Sect(Π
I
/I)
for
the
set
of
sections
of
the
natural
surjection
Π
I
I.
Then
the
following
hold:
the
composite
I
v
→
Π
I
I
[cf.
[NodNon],
(i)
For
any
v
∈
Vert(
G),
Definition
2.2,
(i)]
is
an
isomorphism.
In
particular,
I
v
⊆
Π
I
determines
an
element
s
v
∈
Sect(Π
I
/I);
thus,
we
have
a
map
−→
Sect(Π
I
/I)
Vert(
G)
v
→
s
v
.
Finally,
the
following
equalities
concerning
centralizers
of
sub-
groups
of
Π
I
in
Π
G
[cf.
the
discussion
entitled
“Topological
COMBINATORIAL
ANABELIAN
TOPICS
II
25
groups”
in
“Notations
and
Conventions”]
hold:
Z
Π
G
(s
v
(I))
=
Z
Π
G
(I
v
)
=
Π
v
.
(ii)
The
map
of
(i)
is
injective.
(iii)
If,
moreover,
I
→
Aut(G)
is
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)],
then,
for
any
s
∈
Sect(Π
I
/I),
the
central-
izer
Z
Π
G
(s(I))
is
contained
in
a
verticial
subgroup.
(iv)
Let
s
∈
Sect(Π
I
/I).
Consider
the
following
two
conditions:
(1)
The
section
s
is
contained
in
the
image
of
the
map
of
(i),
i.e.,
s
=
s
v
for
some
v
∈
Vert(
G).
(2)
Z
Π
G
(Z
Π
G
(s(I)))
=
{1}.
Then
we
have
an
implication
(1)
=⇒
(2)
.
If,
moreover,
I
→
Aut(G)
is
of
IPSC-type,
then
we
have
an
equivalence
(1)
⇐⇒
(2)
.
Proof.
First,
we
verify
assertion
(i).
The
fact
that
the
composite
I
v
→
Π
I
I
is
an
isomorphism
follows
from
condition
(2
)
of
[NodNon],
Definition
2.4,
(ii).
On
the
other
hand,
the
equalities
Z
Π
G
(s
v
(I))
=
Z
Π
G
(I
v
)
=
Π
v
follow
from
[NodNon],
Lemma
3.6,
(i).
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
immediately
from
the
final
equalities
of
assertion
(i),
together
with
[NodNon],
Lemma
1.9,
def
(ii).
Next,
we
verify
assertion
(iii).
Write
H
=
Z
Π
G
(s(I)).
Then
it
follows
immediately
from
[CmbGC],
Proposition
2.6,
together
with
the
definition
of
H
=
Z
Π
G
(s(I)),
that
for
any
connected
finite
étale
subcovering
G
→
G
of
G
→
G,
the
image
of
the
composite
H
∩
Π
G
→
Π
G
Π
ab-comb
G
[cf.
the
notation
of
Proposition
1.5,
(iv)]
is
trivial.
Thus,
it
follows
from
the
implication
(iv)
⇒
(i)
of
Proposition
1.5
that
H
is
contained
in
a
verticial
subgroup.
This
completes
the
proof
of
assertion
(iii).
Finally,
we
verify
assertion
(iv).
To
verify
the
implication
(1)
⇒
(2),
suppose
that
condition
(1)
holds.
Then
since
Z
Π
G
(s
v
(I))
=
Z
Π
G
(I
v
)
=
Π
v
[cf.
assertion
(i)]
is
commensurably
terminal
in
Π
G
[cf.
[CmbGC],
Proposition
1.2,
(ii)]
and
center-free
[cf.
[CmbGC],
Remark
1.1.3],
we
conclude
that
Z
Π
G
(Z
Π
G
(s
v
(I)))
=
Z
Π
G
(Π
v
)
=
{1}.
This
completes
the
proof
of
the
implication
(1)
⇒
(2).
Next,
suppose
that
I
→
Aut(G)
is
of
IPSC-type,
and
that
condition
(2)
holds.
Then
it
follows
from
assertion
such
that
H
def
=
Z
Π
G
(s(I))
⊆
(iii)
that
there
exists
a
v
∈
Vert(
G)
Π
v
,
so
I
v
⊆
Z
Π
I
(H).
On
the
other
hand,
since
s(I)
⊆
Z
Π
I
(H),
and
Z
Π
G
(H)
=
Z
Π
G
(Z
Π
G
(s(I)))
=
{1}
[cf.
condition
(2)],
i.e.,
the
composite
of
natural
homomorphisms
Z
Π
I
(H)
→
Π
I
I
is
injective,
it
follows
26
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
that
s(I)
=
Z
Π
I
(H)
⊇
I
v
.
Since
I
v
and
s(I)
may
be
obtained
as
the
images
of
sections,
we
thus
conclude
that
I
v
=
s(I),
i.e.,
s
=
s
v
.
This
completes
the
proof
of
the
implication
(2)
⇒
(1),
hence
also
of
assertion
(iv).
Remark
1.6.1.
Recall
that
in
the
case
of
outer
representations
of
NN-
type,
the
period
matrix
is
not
necessarily
nondegenerate
[cf.
[CbTpI],
Remark
5.9.2].
In
particular,
the
argument
applied
in
the
proof
of
The-
orem
1.6,
(iii)
—
which
depends,
in
an
essential
way,
on
the
fact
that,
in
the
case
of
outer
representations
of
IPSC-type,
the
period
matrix
is
nondegenerate
[cf.
the
proof
of
[CmbGC],
Proposition
2.6]
—
cannot
be
applied
in
the
case
of
outer
representations
of
NN-type.
Nevertheless,
the
question
of
whether
or
not
Theorem
1.6,
(iii),
as
well
as
the
appli-
cation
of
Theorem
1.6,
(iii),
given
in
Corollary
1.7,
(ii),
below,
may
be
generalized
to
the
case
of
outer
representations
of
NN-type
remains
a
topic
of
interest
to
the
authors.
Corollary
1.7
(Group-theoretic
characterization
of
verticial
subgroups
for
outer
representations
of
IPSC-type).
In
the
no-
tation
of
Theorem
1.6,
let
us
refer
to
a
closed
subgroup
of
Π
G
as
a
section-centralizer
if
it
may
be
written
in
the
form
Z
Π
G
(s(I))
for
some
s
∈
Sect(Π
I
/I).
Let
H
⊆
Π
G
be
a
closed
subgroup
of
Π
G
.
Then
the
following
hold:
(i)
Suppose
that
H
is
a
section-centralizer
such
that
Z
Π
G
(H)
=
{1}.
Then
the
following
conditions
on
a
section
s
∈
Sect(Π
I
/I)
are
equivalent:
(i-1)
H
=
Z
Π
G
(s(I)).
(i-2)
s(I)
⊆
Z
Π
I
(H).
(i-3)
s(I)
=
Z
Π
I
(H).
(ii)
Consider
the
following
three
conditions:
(ii-1)
H
is
a
verticial
subgroup.
(ii-2)
H
is
a
section-centralizer
such
that
Z
Π
G
(H)
=
{1}.
(ii-3)
H
is
a
maximal
section-centralizer.
Then
we
have
implications
(ii-1)
=⇒
(ii-2)
=⇒
(ii-3)
.
If,
moreover,
I
→
Aut(G)
is
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)],
then
we
have
equivalences
(ii-1)
⇐⇒
(ii-2)
⇐⇒
(ii-3)
.
COMBINATORIAL
ANABELIAN
TOPICS
II
27
Proof.
First,
we
verify
assertion
(i).
The
implication
(i-1)
⇒
(i-2)
is
immediate.
To
verify
the
implication
(i-2)
⇒
(i-3),
suppose
that
condition
(i-2)
holds.
Then
since
Z
Π
I
(H)
∩
Π
G
=
Z
Π
G
(H)
=
{1},
the
composite
Z
Π
I
(H)
→
Π
I
I
is
injective.
Thus,
since
the
composite
s(I)
→
Z
Π
I
(H)
→
Π
I
I
is
an
isomorphism,
it
follows
immediately
that
condition
(i-3)
holds.
This
completes
the
proof
of
the
implication
(i-2)
⇒
(i-3).
Finally,
to
verify
the
implication
(i-3)
⇒
(i-1),
suppose
that
condition
(i-3)
holds.
Then
since
H
is
a
section-centralizer,
there
exists
a
t
∈
Sect(Π
I
/I)
such
that
H
=
Z
Π
G
(t(I)).
In
particular,
t(I)
⊆
Z
Π
I
(H)
=
s(I)
[cf.
condition
(i-3)].
We
thus
conclude
that
t
=
s,
i.e.,
that
condition
(i-1)
holds.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
The
implication
(ii-1)
⇒
(ii-2)
fol-
lows
immediately
from
Theorem
1.6,
(i),
(iv).
To
verify
the
impli-
cation
(ii-2)
⇒
(ii-3),
suppose
that
H
satisfies
condition
(ii-2);
let
s
∈
Sect(Π
I
/I)
be
such
that
H
⊆
Z
Π
G
(s(I)).
Then
it
follows
imme-
diately
that
s(I)
⊆
Z
Π
I
(H).
Thus,
it
follows
immediately
from
the
equivalence
(i-1)
⇔
(i-2)
of
assertion
(i)
that
H
=
Z
Π
G
(s(I)).
This
completes
the
proof
of
the
implication
(ii-2)
⇒
(ii-3).
Finally,
observe
that
the
implication
(ii-3)
⇒
(ii-1)
in
the
case
where
I
→
Aut(G)
is
of
IPSC-type
follows
immediately
from
Theorem
1.6,
(iii),
together
with
the
fact
that
every
verticial
subgroup
is
a
section-centralizer
[cf.
the
implication
(ii-1)
⇒
(ii-2)
verified
above].
This
completes
the
proof
of
Corollary
1.7.
Lemma
1.8
(Group-theoretic
characterization
of
verticial
sub-
groups
for
outer
representations
of
SNN-type).
Let
H
⊆
Π
G
be
a
closed
subgroup
of
Π
G
and
I
→
Aut(G)
an
outer
representation
of
def
out
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)].
Write
Π
I
=
Π
G
I
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0];
thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Π
G
−→
Π
I
−→
I
−→
1
.
Suppose
that
G
is
untangled
[cf.
[NodNon],
Definition
1.2].
Then
H
is
a
verticial
subgroup
if
and
only
if
H
satisfies
the
following
four
conditions:
def
(i)
The
composite
I
H
=
Z
Π
I
(H)
→
Π
I
I
is
an
isomorphism.
(ii)
It
holds
that
H
=
Z
Π
G
(I
H
).
(iii)
For
any
γ
∈
Π
G
,
it
holds
that
γ
∈
H
if
and
only
if
H
∩
(γ
·
H
·
γ
−1
)
=
{1}.
(iv)
H
contains
a
nontrivial
verticial
element
of
Π
G
[cf.
Defini-
tion
1.1].
28
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Proof.
If
H
is
a
verticial
subgroup,
then
it
is
immediate
that
condition
(iv)
is
satisfied;
moreover,
it
follows
from
condition
(2
)
of
[NodNon],
Definition
2.4,
(ii)
(respectively,
[NodNon],
Lemma
3.6,
(i);
[NodNon],
Remark
1.10.1),
that
H
satisfies
condition
(i)
(respectively,
(ii);
(iii)).
This
completes
the
proof
of
necessity.
To
verify
sufficiency,
suppose
that
H
satisfies
conditions
(i),
(ii),
(iii),
and
(iv).
It
follows
from
condition
(iv)
that
there
exists
a
v
∈
Vert(
G)
def
such
that
J
=
H
∩
Π
v
=
{1}.
If
either
J
=
Π
v
or
J
=
H,
i.e.,
either
Π
v
⊆
H
or
H
⊆
Π
v
,
then
it
is
immediate
that
either
I
H
⊆
I
v
or
I
v
⊆
I
H
[cf.
[NodNon],
Definition
2.2,
(i)].
Thus,
it
follows
from
condition
(i)
[for
H
and
Π
v
]
that
I
H
=
I
v
.
But
then
it
follows
from
condition
(ii)
[for
H
and
Π
v
]
that
H
=
Z
Π
G
(I
H
)
=
Z
Π
G
(I
v
)
=
Π
v
;
in
particular,
H
is
a
verticial
subgroup.
Thus,
we
may
assume
without
loss
of
generality
that
J
=
H,
Π
v
.
def
Let
γ
∈
H
\
J.
Write
J
γ
=
γ
·
J
·
γ
−1
.
Then
we
have
inclusions
Π
v
⊇
J
⊆
H
⊇
J
γ
⊆
Π
v
γ
(=
γ
·
Π
v
·
γ
−1
)
.
Now
we
claim
the
following
assertion:
Claim
1.8.A:
N
Π
G
(J)
=
J,
N
Π
G
(J
γ
)
=
J
γ
.
Indeed,
let
σ
∈
N
Π
G
(J).
Then
since
{1}
=
J
=
J
∩
(σ
·
J
·
σ
−1
)
⊆
Π
v
∩
Π
v
σ
,
it
follows
from
condition
(iii)
[for
Π
v
]
that
σ
∈
Π
v
.
Similarly,
since
{1}
=
J
=
J
∩
(σ
·
J
·
σ
−1
)
⊆
H
∩
(σ
·
H
·
σ
−1
),
it
follows
from
condition
(iii)
[for
H]
that
σ
∈
H.
Thus,
σ
∈
Π
v
∩
H
=
J.
In
particular,
we
obtain
that
N
Π
G
(J)
=
J.
A
similar
argument
implies
that
N
Π
G
(J
γ
)
=
J
γ
.
This
completes
the
proof
of
Claim
1.8.A.
Now
the
composites
N
Π
I
(J),
N
Π
I
(J
γ
)
→
Π
I
I
fit
into
exact
sequences
of
profinite
groups
1
−→
N
Π
G
(J)
−→
N
Π
I
(J)
−→
I
,
1
−→
N
Π
G
(J
γ
)
−→
N
Π
I
(J
γ
)
−→
I
.
Thus,
since
we
have
inclusions
I
H
=
Z
Π
I
(H)
⊆
Z
Π
I
(J)
⊆
N
Π
I
(J)
,
I
H
=
Z
Π
I
(H)
⊆
Z
Π
I
(J
γ
)
⊆
N
Π
I
(J
γ
)
,
I
v
=
Z
Π
I
(Π
v
)
⊆
Z
Π
I
(J)
⊆
N
Π
I
(J)
,
I
v
γ
=
Z
Π
I
(Π
v
γ
)
⊆
Z
Π
I
(J
γ
)
⊆
N
Π
I
(J
γ
)
,
it
follows
immediately
from
Claim
1.8.A,
together
with
condition
(i)
[for
H
and
Π
v
],
that
N
Π
I
(J)
=
J
·
I
H
=
J
·
I
v
,
N
Π
I
(J
γ
)
=
J
γ
·
I
H
=
J
γ
·
I
v
γ
.
In
particular,
we
obtain
that
I
H
⊆
N
Π
I
(J)
=
J
·
I
v
⊆
Π
v
·
D
v
=
D
v
,
I
H
⊆
N
Π
I
(J
γ
)
=
J
γ
·
I
v
γ
⊆
Π
v
γ
·
D
v
γ
=
D
v
γ
COMBINATORIAL
ANABELIAN
TOPICS
II
29
[cf.
[NodNon],
Definition
2.2,
(i)],
i.e.,
I
H
⊆
D
v
∩
D
v
γ
.
On
the
other
hand,
since
H
γ
∈
J
=
H
∩
Π
v
,
it
follows
from
condition
(iii)
[for
Π
v
]
that
Π
v
γ
∩
Π
v
=
{1};
thus,
it
follows
immediately
from
the
fact
that
D
v
∩
D
v
γ
∩
Π
G
=
Π
v
∩
Π
v
γ
=
{1}
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
together
with
condition
(i),
that
I
H
=
D
v
∩
D
v
γ
,
which
implies,
by
[NodNon],
Proposition
3.9,
(iii),
that
there
exists
a
w
∈
Vert(
G)
such
that
I
H
=
I
w
.
In
particular,
it
follows
from
condition
(ii)
[for
H
and
Π
w
]
that
H
=
Z
Π
G
(I
H
)
=
Z
Π
G
(I
w
)
=
Π
w
.
Thus,
H
is
a
verticial
subgroup.
This
completes
the
proof
of
Lemma
1.8.
Theorem
1.9
(Group-theoretic
verticiality/nodality
of
isomor-
phisms
of
outer
representations
of
NN-,
IPSC-type).
Let
Σ
be
a
nonempty
set
of
prime
numbers,
G
(respectively,
H)
a
semi-graph
of
anabelioids
of
pro-Σ
PSC-type,
Π
G
(respectively,
Π
H
)
the
[pro-Σ]
∼
fundamental
group
of
G
(respectively,
H),
α
:
Π
G
→
Π
H
an
isomor-
phism
of
profinite
groups,
I
(respectively,
J)
a
profinite
group,
ρ
I
:
I
→
Aut(G)
(respectively,
ρ
J
:
J
→
Aut(H))
a
continuous
homomorphism,
∼
and
β
:
I
→
J
an
isomorphism
of
profinite
groups.
Suppose
that
the
diagram
I
−−−→
Out(Π
G
)
⏐
⏐
⏐
⏐
Out(α)
β
J
−−−→
Out(Π
H
)
—
where
the
right-hand
vertical
arrow
is
the
isomorphism
induced
by
α;
the
upper
and
lower
horizontal
arrows
are
the
homomorphisms
de-
termined
by
ρ
I
and
ρ
J
,
respectively
—
commutes.
Then
the
following
hold:
(i)
Suppose,
moreover,
that
ρ
I
,
ρ
J
are
of
NN-type
[cf.
[NodNon],
Definition
2.4,
(iii)].
Then
the
following
three
conditions
are
equivalent:
(1)
The
isomorphism
α
is
group-theoretically
verticial
[i.e.,
roughly
speaking,
preserves
verticial
subgroups
—
cf.
[CmbGC],
Definition
1.4,
(iv)].
(2)
The
isomorphism
α
is
group-theoretically
nodal
[i.e.,
roughly
speaking,
preserves
nodal
subgroups
—
cf.
[NodNon],
Definition
1.12].
(3)
There
exists
a
nontrivial
verticial
element
γ
∈
Π
G
such
that
α(γ)
∈
Π
H
is
verticial
[cf.
Definition
1.1].
(ii)
Suppose,
moreover,
that
ρ
I
is
of
NN-type,
and
that
ρ
J
is
of
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)].
[For
example,
this
will
be
the
case
if
both
ρ
I
and
ρ
J
are
of
IPSC-type
—
cf.
30
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[NodNon],
Remark
2.4.2.]
Then
α
is
group-theoretically
verticial,
hence
also
[cf.
(i)]
group-theoretically
nodal.
Proof.
First,
we
verify
assertion
(i).
The
implication
(1)
⇒
(2)
fol-
lows
from
[NodNon],
Proposition
1.13.
The
implication
(2)
⇒
(3)
follows
from
the
fact
that
any
nodal
subgroup
is
contained
in
a
verti-
cial
subgroup.
[Note
that
if
Node(H)
=
∅,
then
every
element
of
Π
H
is
verticial.]
Finally,
we
verify
the
implication
(3)
⇒
(1).
Suppose
that
condition
(3)
holds.
Since
verticial
subgroups
are
commensurably
terminal
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
to
verify
the
implica-
tion
(3)
⇒
(1),
by
replacing
Π
I
,
Π
J
by
open
subgroups
of
Π
I
,
Π
J
,
we
may
assume
without
loss
of
generality
that
ρ
I
,
ρ
J
are
of
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)],
and,
moreover,
that
G
and
H
are
un-
tangled
[cf.
[NodNon],
Definition
1.2;
[NodNon],
Remark
1.2.1,
(i),
(ii)].
be
such
that
γ
∈
Π
v
.
Then
it
is
immediate
that
α(Π
v
)
Let
v
∈
Vert(
G)
satisfies
conditions
(i),
(ii),
and
(iii)
in
the
statement
of
Lemma
1.8.
On
the
other
hand,
it
follows
from
condition
(3)
that
α(Π
v
)
satisfies
condition
(iv)
in
the
statement
of
Lemma
1.8.
Thus,
it
follows
from
Lemma
1.8
that
α(Π
v
)
⊆
Π
H
is
a
verticial
subgroup.
Now
it
follows
from
[NodNon],
Theorem
4.1,
that
α
is
group-theoretically
verticial.
This
completes
the
proof
of
the
implication
(3)
⇒
(1).
Finally,
we
verify
assertion
(ii).
It
is
immediate
that,
to
verify
as-
sertion
(ii)
—
by
replacing
I,
J
by
open
subgroups
of
I,
J
—
we
may
assume
without
loss
of
generality
that
ρ
I
is
of
SNN-type.
Let
H
⊆
Π
G
be
a
verticial
subgroup
of
Π
G
.
Then
it
follows
from
Corollary
1.7,
(ii),
that
H,
hence
also
α(H),
is
a
maximal
section-centralizer
[cf.
the
statement
of
Corollary
1.7].
Thus,
since
ρ
J
is
of
IPSC-type,
again
by
Corollary
1.7,
(ii),
we
conclude
that
α(H)
⊆
Π
H
is
a
verticial
subgroup
of
Π
H
.
In
particular,
it
follows
from
[NodNon],
Theorem
4.1,
together
with
[NodNon],
Remark
2.4.2,
that
α
is
group-theoretically
verticial
and
group-theoretically
nodal.
This
completes
the
proof
of
assertion
(ii).
Remark
1.9.1.
Thus,
Theorem
1.9,
(i),
may
be
regarded
as
a
gen-
eralization
of
[NodNon],
Corollary
4.2.
Of
course,
ideally,
one
would
like
to
be
able
to
prove
that
conditions
(1)
and
(2)
of
Theorem
1.9,
(i),
hold
automatically
[i.e.,
as
in
the
case
of
outer
representations
of
IPSC-type
treated
in
Theorem
1.9,
(ii)],
without
assuming
condition
(3).
Although
this
topic
lies
beyond
the
scope
of
the
present
mono-
graph,
perhaps
progress
could
be
made
in
this
direction
if,
say,
in
the
case
where
Σ
is
either
equal
to
the
set
of
all
prime
numbers
or
of
car-
dinality
one,
one
starts
with
an
isomorphism
α
that
arises
from
a
PF-
admissible
[cf.
[CbTpI],
Definition
1.4,
(i)]
isomorphism
between
con-
figuration
space
groups
corresponding
to
m-dimensional
configuration
spaces
[where
m
≥
2]
associated
to
stable
curves
that
give
rise
to
G
and
COMBINATORIAL
ANABELIAN
TOPICS
II
31
H,
respectively
[i.e.,
one
assumes
the
condition
of
“m-cuspidalizability”
discussed
in
Definition
3.20,
below,
where
we
replace
the
condition
of
“PFC-admissibility”
by
the
condition
of
“PF-admissibility”].
For
in-
stance,
if
Cusp(G)
=
∅,
then
it
follows
from
[CbTpI],
Theorem
1.8,
(iv);
[NodNon],
Corollary
4.2,
that
this
condition
on
α
is
sufficient
to
imply
that
conditions
(1)
and
(2)
of
Theorem
1.9,
(i),
hold.
32
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
2.
Partial
combinatorial
cuspidalization
for
F-admissible
outomorphisms
In
the
present
§2,
we
apply
the
results
obtained
in
the
preceding
§1,
together
with
the
theory
developed
by
the
authors
in
earlier
papers,
to
prove
combinatorial
cuspidalization-type
results
for
F-admissible
out-
omorphisms
[cf.
Theorem
2.3,
(i),
below].
We
also
show
that
any
F-
admissible
outomorphism
of
a
configuration
space
group
[arising
from
a
configuration
space]
of
sufficiently
high
dimension
[i.e.,
≥
3
in
the
affine
case;
≥
4
in
the
proper
case]
is
necessarily
C-admissible,
i.e.,
preserves
the
cuspidal
inertia
subgroups
of
the
various
subquotients
correspond-
ing
to
surface
groups
[cf.
Theorem
2.3,
(ii),
below].
Finally,
we
discuss
applications
of
these
combinatorial
anabelian
results
to
the
anabelian
geometry
of
configuration
spaces
associated
to
hyperbolic
curves
over
arithmetic
fields
[cf.
Corollaries
2.5,
2.6,
below].
In
the
present
§2,
let
Σ
be
a
set
of
prime
numbers
which
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardinality
one;
n
a
positive
integer;
k
an
algebraically
closed
field
of
characteristic
∈
Σ;
X
a
hyper-
bolic
curve
of
type
(g,
r)
over
k.
For
each
positive
integer
i,
write
X
i
for
the
i-th
configuration
space
of
X;
Π
i
for
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
X
i
.
Definition
2.1.
Let
α
∈
Aut(Π
n
)
be
an
automorphism
of
Π
n
.
(i)
Write
{1}
=
K
n
⊆
K
n−1
⊆
·
·
·
⊆
K
2
⊆
K
1
⊆
K
0
=
Π
n
for
the
standard
fiber
filtration
on
Π
n
[cf.
[CmbCsp],
Defini-
tion
1.1,
(i)].
For
each
m
∈
{1,
2,
·
·
·
,
n},
write
C
m
for
the
[finite]
set
of
K
m−1
/K
m
-conjugacy
classes
of
cuspidal
inertia
subgroups
of
K
m−1
/K
m
[where
we
recall
that
K
m−1
/K
m
is
equipped
with
a
natural
structure
of
pro-Σ
surface
group
—
cf.
[MzTa],
Definition
1.2].
Then
we
shall
say
that
α
is
wC-
admissible
[i.e.,
“weakly
C-admissible”]
if
α
preserves
the
stan-
dard
fiber
filtration
on
Π
n
and,
moreover,
satisfies
the
following
conditions:
•
If
m
∈
{1,
2,
·
·
·
n−1},
then
the
automorphism
of
K
m−1
/K
m
determined
by
α
induces
an
automorphism
of
C
m
.
•
It
follows
immediately
from
the
various
definitions
in-
volved
that
we
have
a
natural
injection
C
n−1
→
C
n
.
That
is
to
say,
if
one
thinks
of
K
n−2
as
the
two-dimensional
configuration
space
group
associated
to
some
hyperbolic
curve,
then
the
image
of
C
n−1
→
C
n
corresponds
to
the
set
of
cusps
of
a
fiber
[of
the
two-dimensional
configura-
tion
space
over
the
hyperbolic
curve]
that
arise
from
the
COMBINATORIAL
ANABELIAN
TOPICS
II
33
cusps
of
the
hyperbolic
curve.
Then
the
automorphism
of
K
n−1
determined
by
α
induces
an
automorphism
of
the
image
of
the
natural
injection
C
n−1
→
C
n
.
Write
Aut
wC
(Π
n
)
⊆
Aut(Π
n
)
for
the
subgroup
of
wC-admissible
automorphisms
and
def
Out
wC
(Π
n
)
=
Aut
wC
(Π
n
)/Inn(Π
n
)
⊆
Out(Π
n
)
.
We
shall
refer
to
an
element
of
Out
wC
(Π
n
)
as
a
wC-admissible
outomorphism.
(ii)
We
shall
say
that
α
is
FwC-admissible
if
α
is
F-admissible
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
and
wC-admissible
[cf.
(i)].
Write
Aut
FwC
(Π
n
)
⊆
Aut
F
(Π
n
)
for
the
subgroup
of
FwC-admissible
automorphisms
and
def
Out
FwC
(Π
n
)
=
Aut
FwC
(Π
n
)/Inn(Π
n
)
⊆
Out
F
(Π
n
)
.
We
shall
refer
to
an
element
of
Out
FwC
(Π
n
)
as
an
FwC-admissible
outomorphism.
(iii)
We
shall
say
that
α
is
DF-admissible
[i.e.,
“diagonal-fiber-
admissible”]
if
α
is
F-admissible,
and,
moreover,
α
induces
the
same
automorphism
of
Π
1
relative
to
the
various
quotients
Π
n
Π
1
by
fiber
subgroups
of
co-length
1
[cf.
[MzTa],
Defini-
tion
2.3,
(iii)].
Write
Aut
DF
(Π
n
)
⊆
Aut
F
(Π
n
)
for
the
subgroup
of
DF-admissible
automorphisms.
Remark
2.1.1.
Thus,
it
follows
immediately
from
the
definitions
that
C-admissible
=⇒
wC-admissible.
In
particular,
we
have
inclusions
Aut
FC
(Π
n
)
⊂
Aut
FwC
(Π
n
)
∩
Aut
C
(Π
n
)
⊂
Out
FC
(Π
n
)
⊂
Out
FwC
(Π
n
)
∩
∩
Aut
wC
(Π
n
)
Out
C
(Π
n
)
[cf.
Definition
2.1,
(i),
(ii)].
∩
⊂
Out
wC
(Π
n
)
34
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Lemma
2.2
(F-admissible
automorphisms
and
inertia
subgroups).
Let
α
∈
Aut
F
(Π
n
)
be
an
F-admissible
automorphism
of
Π
n
.
Then
the
following
hold:
(i)
There
exist
β
∈
Aut
DF
(Π
n
)
[cf.
Definition
2.1,
(iii)]
and
ι
∈
Inn(Π
n
)
such
that
α
=
β
◦
ι.
(ii)
For
each
positive
integer
i,
write
Z
i
log
for
the
i-th
log
config-
uration
space
of
X
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”];
U
Z
i
⊆
Z
i
for
the
interior
of
Z
i
log
[cf.
the
discussion
entitled
“Log
schemes”
in
[CbTpI],
§0],
which
may
be
identified
with
X
i
.
Let
be
an
irreducible
component
of
the
complement
Z
n−1
\
U
Z
n−1
[cf.
[CmbCsp],
Proposition
1.3];
I
⊆
Π
n−1
an
inertia
subgroup
of
Π
n−1
asso-
ciated
to
the
divisor
of
Z
n−1
;
pr
:
U
Z
n
→
U
Z
n−1
the
projection
obtained
by
forgetting
the
factor
labeled
n;
pr
Π
:
Π
n
Π
n−1
def
the
surjection
induced
by
pr;
Π
n/n−1
=
Ker(pr
Π
);
θ
an
ir-
reducible
component
of
the
fiber
of
the
[uniquely
determined]
extension
Z
n
→
Z
n−1
of
pr
over
the
generic
point
of
[so
θ
naturally
determines
an
irreducible
component
of
the
com-
plement
Z
n
\
U
Z
n
];
D
I
θ
⊆
Π
n
×
Π
n−1
I
(⊆
Π
n
)
—
where
the
homomorphism
Π
n
→
Π
n−1
implicit
in
the
fiber
product
is
the
surjection
pr
Π
:
Π
n
Π
n−1
—
a
decomposition
subgroup
of
Π
n
×
Π
n−1
I
(⊆
Π
n
)
associated
to
the
divisor
[naturally
deter-
def
mined
by]
θ
of
Z
n
;
Π
θ
=
D
I
θ
∩
Π
n/n−1
[cf.
[CmbCsp],
Proposi-
tion
1.3,
(iv)].
Suppose
that
the
automorphism
of
Π
n−1
induced
by
α
∈
Aut
F
(Π
n
)
relative
to
pr
Π
stabilizes
I
⊆
Π
n−1
.
Then
α
preserves
the
Π
n/n−1
-conjugacy
class
of
Π
θ
.
Proof.
Assertion
(i)
follows
immediately
from
[CbTpI],
Theorem
A,
(i).
Assertion
(ii)
follows
immediately
from
Theorem
1.9,
(ii)
[cf.
also
the
proof
of
[CmbCsp],
Proposition
1.3,
(iv)].
Theorem
2.3
(Partial
combinatorial
cuspidalization
for
F-ad-
missible
outomorphisms).
Let
Σ
be
a
set
of
prime
numbers
which
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardinality
one;
n
a
positive
integer;
X
a
hyperbolic
curve
of
type
(g,
r)
over
an
alge-
braically
closed
field
of
characteristic
∈
Σ;
X
n
the
n-th
configuration
space
of
X;
Π
n
the
maximal
pro-Σ
quotient
of
the
fundamental
group
of
X
n
;
Out
F
(Π
n
)
⊆
Out(Π
n
)
the
subgroup
of
F-admissible
outomorphisms
[i.e.,
roughly
speaking,
outomorphisms
that
preserve
the
fiber
subgroups
—
cf.
[CmbCsp],
Def-
inition
1.1,
(ii)]
of
Π
n
;
Out
FC
(Π
n
)
⊆
Out
F
(Π
n
)
COMBINATORIAL
ANABELIAN
TOPICS
II
35
the
subgroup
of
FC-admissible
outomorphisms
[i.e.,
roughly
speak-
ing,
outomorphisms
that
preserve
the
fiber
subgroups
and
the
cuspidal
inertia
subgroups
—
cf.
[CmbCsp],
Definition
1.1,
(ii)]
of
Π
n
;
(Out
FC
(Π
n
)
⊆)
Out
FwC
(Π
n
)
⊆
Out
F
(Π
n
)
the
subgroup
of
FwC-admissible
outomorphisms
[cf.
Definition
2.1,
(ii);
Remark
2.1.1]
of
Π
n
.
Then
the
following
hold:
(i)
Write
def
n
inj
=
1
2
if
r
=
0,
if
r
=
0
,
def
n
bij
=
3
4
if
r
=
0,
if
r
=
0
.
If
n
≥
n
inj
(respectively,
n
≥
n
bij
),
then
the
natural
homomor-
phism
Out
F
(Π
n+1
)
−→
Out
F
(Π
n
)
induced
by
the
projections
X
n+1
→
X
n
obtained
by
forgetting
any
one
of
the
n
+
1
factors
of
X
n+1
[cf.
[CbTpI],
Theorem
A,
(i)]
is
injective
(respectively,
bijective).
(ii)
Write
⎧
⎨
2
def
3
n
FC
=
⎩
4
if
(g,
r)
=
(0,
3),
if
(g,
r)
=
(0,
3)
and
r
=
0,
if
r
=
0
.
If
n
≥
n
FC
,
then
it
holds
that
Out
FC
(Π
n
)
=
Out
F
(Π
n
)
.
(iii)
Write
⎧
⎨
2
def
3
n
FwC
=
⎩
4
if
r
≥
2,
if
r
=
1,
if
r
=
0
.
If
n
≥
n
FwC
,
then
it
holds
that
Out
FwC
(Π
n
)
=
Out
F
(Π
n
)
.
(iv)
Consider
the
natural
inclusion
S
n
→
Out(Π
n
)
—
where
we
write
S
n
for
the
symmetric
group
on
n
letters
—
obtained
by
permuting
the
various
factors
of
X
n
.
If
(r,
n)
=
(0,
2),
then
the
image
of
this
inclusion
is
contained
in
the
cen-
tralizer
Z
Out(Π
n
)
(Out
F
(Π
n
)).
Proof.
First,
we
verify
assertion
(iii)
in
the
case
where
n
=
2,
which
implies
that
r
≥
2
[cf.
the
statement
of
assertion
(iii)].
To
verify
36
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
assertion
(iii)
in
the
case
where
n
=
2,
it
is
immediate
that
it
suffices
to
verify
that
Aut
FwC
(Π
2
)
=
Aut
F
(Π
2
)
.
Let
α
∈
Aut
F
(Π
2
).
Let
us
assign
the
cusps
of
X
the
labels
a
1
,
·
·
·
,
a
r
.
Now,
for
each
i
∈
{1,
·
·
·
,
r},
recall
that
there
is
a
uniquely
determined
cusp
of
the
geometric
generic
fiber
X
2/1
of
the
projection
X
2
→
X
to
the
factor
labeled
1
that
corresponds
naturally
to
the
cusp
of
X
labeled
a
i
;
we
assign
to
this
uniquely
determined
cusp
the
label
b
i
.
Thus,
there
is
precisely
one
cusp
of
X
2/1
that
has
not
been
assigned
a
label
∈
{b
1
,
·
·
·
,
b
r
};
we
assign
to
this
uniquely
determined
cusp
the
label
b
r+1
.
Then
since
the
automorphism
of
Π
1
induced
by
α
relative
to
either
p
1
or
p
2
—
where
we
write
p
1
,
p
2
for
the
surjections
Π
2
Π
1
induced
by
the
projections
X
2
→
X
to
the
factors
labeled
1,
2,
respectively
—
is
FC-admissible
[cf.
[CbTpI],
Theorem
A,
(ii)],
it
follows
from
the
various
definitions
involved
that,
to
verify
that
α
∈
Aut
FwC
(Π
2
),
it
suffices
to
verify
the
following
assertion:
def
Claim
2.3.A:
For
any
b
∈
{b
1
,
·
·
·
,
b
r
},
if
I
b
⊆
Π
2/1
=
Ker(p
1
)
⊆
Π
2
is
a
cuspidal
inertia
subgroup
associated
to
the
cusp
labeled
b,
then
α(I
b
)
is
a
cuspidal
inertia
subgroup.
Now
observe
that
to
verify
Claim
2.3.A,
by
replacing
α
by
the
compos-
ite
of
α
with
a
suitable
element
of
Aut
FC
(Π
2
)
[cf.
[CmbCsp],
Lemma
2.4],
we
may
assume
without
loss
of
generality
that
the
[necessarily
FC-
admissible]
automorphism
of
Π
1
induced
by
α
relative
to
p
1
,
hence
also
relative
to
p
2
[cf.
[CbTpI],
Theorem
A,
(i)],
induces
the
identity
auto-
morphism
on
the
set
of
conjugacy
classes
of
cuspidal
inertia
subgroups
of
Π
1
.
To
verify
Claim
2.3.A,
let
us
fix
b
∈
{b
1
,
·
·
·
,
b
r
},
together
with
a
cuspidal
inertia
subgroup
I
b
⊆
Π
2/1
associated
to
the
cusp
labeled
b
of
Π
2/1
.
Also,
let
us
fix
•
a
∈
{a
1
,
·
·
·
,
a
r
}
such
that
if
b
=
b
i
and
a
=
a
j
,
then
i
=
j
[cf.
the
assumption
that
r
≥
2!];
•
a
cuspidal
inertia
subgroup
I
a
⊆
Π
1
associated
to
the
cusp
labeled
a
of
Π
1
.
Now
observe
that
since
the
[necessarily
FC-admissible]
automorphism
of
Π
1
induced
by
α
relative
to
p
1
induces
the
identity
automorphism
on
the
set
of
conjugacy
classes
of
cuspidal
inertia
subgroups
of
Π
1
,
to
verify
the
fact
that
α(I
b
)
is
a
cuspidal
inertia
subgroup,
we
may
assume
without
loss
of
generality
[by
replacing
α
by
a
suitable
Π
2
-conjugate
of
α]
that
the
automorphism
of
Π
1
induced
by
α
relative
to
p
1
fixes
I
a
.
Let
Π
F
a
⊆
Π
2/1
be
a
major
verticial
subgroup
at
a
[cf.
[CmbCsp],
Definition
1.4,
(ii)]
such
that
I
b
⊆
Π
F
a
.
Then
it
follows
from
Lemma
2.2,
(ii),
COMBINATORIAL
ANABELIAN
TOPICS
II
37
that
α
fixes
the
Π
2/1
-conjugacy
class
of
Π
F
a
,
i.e.,
that
Π
†
F
a
=
α(Π
F
a
)
is
a
Π
2/1
-conjugate
of
Π
F
a
.
Thus,
one
verifies
easily
that,
to
verify
that
α(I
b
)
is
a
cuspidal
inertia
subgroup,
it
suffices
to
verify
that
the
∼
isomorphism
Π
F
a
→
Π
†
F
a
induced
by
α
is
group-theoretically
cuspidal
—
cf.
[CmbGC],
Definition
1.4,
(iv).
[Note
that
it
follows
immediately
from
the
various
definitions
involved
that
Π
F
a
and
Π
†
F
a
may
be
regarded
as
pro-Σ
fundamental
groups
of
semi-graphs
of
anabelioids
of
pro-Σ
PSC-type.]
On
the
other
hand,
it
follows
immediately
from
the
various
definitions
involved
that
this
isomorphism
factors
as
the
composite
def
∼
∼
∼
Π
F
a
−→
Π
1
−→
Π
1
←−
Π
†
F
a
—
where
the
first
and
third
arrows
are
the
isomorphisms
induced
by
p
2
:
Π
2
Π
1
[cf.
[CmbCsp],
Definition
1.4,
(ii)],
and
the
second
ar-
row
is
the
automorphism
induced
by
α
relative
to
p
2
—
and
that
the
three
arrows
appearing
in
this
composite
are
group-theoretically
cuspi-
dal.
Thus,
we
conclude
that
α(I
b
)
is
a
cuspidal
inertia
subgroup.
This
completes
the
proof
of
Claim
2.3.A,
hence
also
of
assertion
(iii)
in
the
case
where
n
=
2.
Next,
we
verify
assertion
(ii)
in
the
case
where
(g,
r,
n)
=
(0,
3,
2).
In
the
following,
we
shall
use
the
notation
“a
i
”
[for
i
=
1,
2,
3]
and
“b
j
”
[for
j
=
1,
2,
3,
4]
introduced
in
the
proof
of
assertion
(iii)
in
the
case
where
n
=
2.
Now,
to
verify
assertion
(ii)
in
the
case
where
(g,
r,
n)
=
(0,
3,
2),
it
is
immediate
that
it
suffices
to
verify
that
Aut
FC
(Π
2
)
=
Aut
F
(Π
2
)
.
Let
α
∈
Aut
F
(Π
2
).
Then
let
us
observe
that
to
verify
that
α
∈
Aut
FC
(Π
2
),
by
replacing
α
by
the
composite
of
α
with
a
suitable
ele-
ment
of
Aut
FC
(Π
2
)
[cf.
[CmbCsp],
Lemma
2.4],
we
may
assume
without
loss
of
generality
that
the
[necessarily
FC-admissible
—
cf.
[CbTpI],
Theorem
A,
(ii)]
automorphism
of
Π
1
induced
by
α
relative
to
p
1
,
hence
also
relative
to
p
2
[cf.
[CbTpI],
Theorem
A,
(i)]
—
where
we
write
p
1
,
p
2
for
the
surjections
Π
2
Π
1
induced
by
the
projections
X
2
→
X
to
the
factors
labeled
1,
2,
respectively
—
induces
the
identity
automor-
phism
on
the
set
of
conjugacy
classes
of
cuspidal
inertia
subgroups
of
Π
1
.
Now
it
follows
from
assertion
(iii)
in
the
case
where
n
=
2
that
α
is
FwC-admissible;
thus,
to
verify
the
fact
that
α
is
FC-admissible,
it
suffices
to
verify
the
following
assertion:
def
Claim
2.3.B:
If
I
b
4
⊆
Π
2/1
=
Ker(p
1
)
⊆
Π
2
is
a
cuspi-
dal
inertia
subgroup
associated
to
the
cusp
labeled
b
4
,
then
α(I
b
4
)
is
a
cuspidal
inertia
subgroup.
On
the
other
hand,
as
is
well-known
[cf.
e.g.,
[CbTpI],
Lemma
6.10,
(ii)],
there
exists
an
automorphism
of
X
2
over
X
relative
to
the
projec-
tion
to
the
factor
labeled
1
which
switches
the
cusps
on
the
geometric
38
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
generic
fiber
X
2/1
labeled
b
1
and
b
4
.
In
particular,
there
exists
an
auto-
morphism
ι
of
Π
2
over
Π
1
relative
to
p
1
which
switches
the
respective
Π
2/1
-conjugacy
classes
of
cuspidal
inertia
subgroups
associated
to
b
1
and
b
4
.
Write
β
=
ι
−1
◦
α
◦
ι.
Now
let
us
verify
that
Claim
2.3.B
follows
from
the
following
asser-
tion:
Claim
2.3.C:
β
∈
Aut
F
(Π
2
).
Indeed,
if
Claim
2.3.C
holds,
then
it
follows
from
assertion
(iii)
in
the
case
where
n
=
2
that,
for
any
cuspidal
inertia
subgroup
I
b
1
⊆
Π
2/1
associated
to
the
cusp
labeled
b
1
,
β(I
b
1
)
is
a
cuspidal
inertia
subgroup.
Thus,
it
follows
immediately
from
our
choice
of
ι
that,
for
any
cuspidal
inertia
subgroup
I
b
4
⊆
Π
2/1
associated
to
the
cusp
labeled
b
4
,
α(I
b
4
)
is
a
cuspidal
inertia
subgroup.
This
completes
the
proof
of
the
assertion
that
Claim
2.3.C
implies
Claim
2.3.B.
Finally,
we
verify
Claim
2.3.C.
Since
α
and
ι,
hence
also
β,
preserve
Π
2/1
⊆
Π
2
,
it
follows
immediately
from
[CmbCsp],
Proposition
1.2,
(i),
that,
to
verify
Claim
2.3.C,
it
suffices
to
verify
that
β
preserves
Ξ
2
⊆
Π
2
[cf.
[CmbCsp],
Definition
1.1,
(iii)],
i.e.,
the
normal
closed
subgroup
of
Π
2
topologically
normally
generated
by
a
cuspidal
inertia
subgroup
associated
to
b
4
.
On
the
other
hand,
this
follows
immediately
from
the
fact
that
α
preserves
the
Π
2/1
-conjugacy
class
of
cuspidal
inertia
subgroups
associated
to
b
1
[cf.
assertion
(iii)
in
the
case
where
n
=
2],
together
with
our
choice
of
ι.
This
completes
the
proof
of
Claim
2.3.C,
hence
also
of
assertion
(ii)
in
the
case
where
(g,
r,
n)
=
(0,
3,
2).
Next,
we
verify
assertion
(ii)
in
the
case
where
(g,
r,
n)
=
(0,
3,
2).
Thus,
n
≥
3.
Write
Π
†
3
(respectively,
Π
†
2
;
Π
†
1
)
for
the
kernel
of
the
surjection
Π
n
Π
n−3
(respectively,
Π
n−1
Π
n−3
;
Π
n−2
Π
n−3
)
induced
by
the
projection
obtained
by
forgetting
the
factor(s)
labeled
n,
n
−
1,
n
−
2
(respectively,
n
−
1,
n
−
2;
n
−
2).
Here,
if
n
=
3,
then
def
we
set
Π
n−3
=
Π
0
=
{1}.
Then
recall
[cf.,
e.g.,
the
proof
of
[CmbCsp],
Theorem
4.1,
(i)]
that
we
have
natural
isomorphisms
out
out
out
Π
n
≃
Π
†
3
Π
n−3
;
Π
n−1
≃
Π
†
2
Π
n−3
;
Π
n−2
≃
Π
†
1
Π
n−3
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
Also,
we
recall
[cf.
[MzTa],
Proposition
2.4,
(i)]
that
one
may
interpret
the
surjections
Π
†
3
Π
†
2
Π
†
1
induced
by
the
surjections
Π
n
Π
n−1
Π
n−2
as
the
surjections
“Π
3
Π
2
Π
1
”
that
arise
from
the
projec-
tions
X
3
→
X
2
→
X
in
the
case
of
an
“X”
of
type
(g,
r
+
n
−
3).
Moreover,
one
verifies
easily
that
this
interpretation
is
compatible
with
the
definition
of
the
various
“Out(−)’s”
involved.
Thus,
since
n
FC
=
4
if
r
=
0,
the
above
natural
isomorphisms,
together
with
[CbTpI],
The-
orem
A,
(ii),
allow
one
to
reduce
the
equality
in
question
to
the
case
where
n
=
3
and
r
=
0.
COMBINATORIAL
ANABELIAN
TOPICS
II
39
Now
one
verifies
easily
that,
to
verify
the
equality
in
question
in
the
case
where
n
=
3
and
r
=
0,
it
is
immediate
that
it
suffices
to
verify
that
Aut
FC
(Π
3
)
=
Aut
F
(Π
3
)
.
Let
α
∈
Aut
F
(Π
3
).
Then
let
us
observe
that
to
verify
α
∈
Aut
FC
(Π
3
),
by
replacing
α
by
the
composite
of
α
with
a
suitable
element
of
Aut
FC
(Π
3
)
[cf.
[CmbCsp],
Lemma
2.4],
we
may
assume
without
loss
of
general-
ity
that
the
[necessarily
FC-admissible
—
cf.
[CbTpI],
Theorem
A,
(ii)]
automorphism
of
Π
1
induced
by
α
relative
to
q
1
,
hence
also
rel-
ative
to
either
q
2
or
q
3
[cf.
[CbTpI],
Theorem
A,
(i)]
—
where
we
write
q
1
,
q
2
,
q
3
for
the
surjections
Π
3
Π
1
induced
by
the
projections
X
3
→
X
to
the
factors
labeled
1,
2,
3,
respectively
—
induces
the
iden-
tity
automorphism
on
the
set
of
conjugacy
classes
of
cuspidal
inertia
subgroups
of
Π
1
;
in
particular,
one
verifies
easily
that
the
[necessarily
FC-admissible
—
cf.
[CbTpI],
Theorem
A,
(ii)]
automorphism
of
Π
2/1
—
where
we
write
p
1
:
Π
2
Π
1
for
the
surjection
induced
by
the
pro-
def
jection
X
2
→
X
to
the
factor
labeled
1
and
Π
2/1
=
Ker(p
1
)
⊆
Π
2
—
induced
by
α
induces
the
identity
automorphism
on
the
set
of
conjugacy
classes
of
cuspidal
inertia
subgroups
of
Π
2/1
.
Write
X
2/1
(respectively,
X
3/2
;
X
3/1
)
for
the
geometric
generic
fiber
of
the
projection
X
2
→
X
(respectively,
X
3
→
X
2
;
X
3
→
X)
to
the
factor(s)
labeled
1
(respec-
tively,
1,
2;
1).
Let
us
assign
the
cusps
of
X
the
labels
a
1
,
·
·
·
,
a
r
.
For
each
i
∈
{1,
·
·
·
,
r},
we
assign
to
the
cusp
of
X
2/1
that
corresponds
nat-
urally
to
the
cusp
of
X
labeled
a
i
the
label
b
i
.
Thus,
there
is
precisely
one
cusp
of
X
2/1
that
has
not
been
assigned
a
label
∈
{b
1
,
·
·
·
,
b
r
};
we
assign
to
this
uniquely
determined
cusp
the
label
b
r+1
.
For
each
i
∈
{1,
·
·
·
,
r
+
1},
we
assign
to
the
cusp
of
X
3/2
that
corresponds
nat-
urally
to
the
cusp
of
X
2/1
labeled
b
i
the
label
c
i
.
Thus,
there
is
precisely
one
cusp
of
X
3/2
that
has
not
been
assigned
a
label
∈
{c
1
,
·
·
·
,
c
r+1
};
we
assign
to
this
uniquely
determined
cusp
the
label
c
r+2
.
Now
it
follows
from
assertion
(iii)
in
the
case
where
n
=
2,
applied
to
the
restriction
of
def
α
to
Π
3/1
=
Ker(q
1
),
together
with
[CbTpI],
Theorem
A,
(ii),
that
α
is
FwC-admissible.
Write
q
12
:
Π
3
Π
2
for
the
surjection
induced
by
the
def
projection
X
3
→
X
2
to
the
factors
labeled
1,
2;
Π
3/2
=
Ker(q
12
)
⊆
Π
3
.
Thus,
to
verify
the
fact
that
α
is
FC-admissible,
it
suffices
to
verify
the
following
assertion:
Claim
2.3.D:
If
I
c
r+2
⊆
Π
3/2
is
a
cuspidal
inertia
sub-
group
associated
to
the
cusp
labeled
c
r+2
,
then
α(I
c
r+2
)
is
a
cuspidal
inertia
subgroup.
To
verify
Claim
2.3.D,
let
us
fix
a
cusp
labeled
b
∈
{b
1
,
·
·
·
,
b
r
}
[where
we
recall
that
r
=
0],
a
cuspidal
inertia
subgroup
I
b
⊆
Π
2/1
as-
sociated
to
the
cusp
labeled
b
of
X
2/1
,
and
a
cuspidal
inertia
subgroup
I
c
r+2
⊆
Π
3/2
associated
to
the
cusp
labeled
c
r+2
of
Π
3/2
.
Now
observe
40
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
that
since
the
[necessarily
FC-admissible]
automorphism
of
Π
2/1
in-
duced
by
α
induces
the
identity
automorphism
on
the
set
of
conjugacy
classes
of
cuspidal
inertia
subgroups
of
Π
2/1
,
to
verify
the
assertion
that
α(I
c
r+2
)
is
a
cuspidal
inertia
subgroup,
we
may
assume
without
loss
of
generality
[by
replacing
α
by
a
suitable
Π
3
-conjugate
of
α]
that
the
automorphism
of
Π
2/1
induced
by
α
fixes
I
b
.
Let
Π
E
b
⊆
Π
3/2
be
a
minor
verticial
subgroup,
relative
to
the
two-dimensional
configuration
space
X
3/1
associated
to
the
hyperbolic
curve
X
2/1
,
at
the
cusp
labeled
b
[cf.
[CmbCsp],
Definition
1.4,
(ii)]
such
that
I
c
r+2
⊆
Π
E
b
.
Then
it
fol-
lows
immediately
from
Lemma
2.2,
(ii),
that
α
fixes
the
Π
3/2
-conjugacy
class
of
Π
E
b
,
i.e.,
that
Π
†
E
b
=
α(Π
E
b
)
is
a
Π
3/2
-conjugate
of
Π
E
b
.
Thus,
one
verifies
easily
that,
to
verify
that
α(I
c
r+2
)
is
a
cuspidal
inertia
sub-
∼
group,
it
suffices
to
verify
that
the
isomorphism
Π
E
b
→
Π
†
E
b
induced
by
α
is
group-theoretically
cuspidal
—
cf.
[CmbGC],
Definition
1.4,
(iv).
[Note
that
it
follows
immediately
from
the
various
definitions
involved
that
Π
E
b
and
Π
†
E
b
may
be
regarded
as
pro-Σ
fundamental
groups
of
semi-graphs
of
anabelioids
of
pro-Σ
PSC-type.]
On
the
other
hand,
it
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
discussion
concerning
the
isomorphism
of
the
second
display
of
[CmbCsp],
Definition
1.4,
(ii),
that
the
composites
def
Π
E
b
,
Π
†
E
b
→
Π
3/2
Π
2/1
—
where
the
second
arrow
is
the
surjection
determined
by
the
surjec-
tion
q
13
:
Π
3
Π
2
induced
by
the
projection
X
3
→
X
2
to
the
factors
labeled
1,
3
—
are
injective,
and
that
the
Π
2/1
-conjugacy
class
of
the
image
in
Π
2/1
of
either
of
these
composite
injections
coincides
with
the
Π
2/1
-conjugacy
class
of
a
minor
verticial
subgroup
at
the
cusp
labeled
a
i
[where
we
write
b
=
b
i
—
cf.
[CmbCsp],
Definition
1.4,
(ii)].
In
particular,
since
the
automorphism
of
Π
2
induced
by
α
relative
to
q
13
is
FC-admissible
[cf.
[CbTpI],
Theorem
A,
(ii)],
it
follows
immediately
∼
that
the
isomorphism
Π
E
b
→
Π
†
E
b
induced
by
α
is
group-theoretically
cuspidal.
This
completes
the
proof
of
Claim
2.3.D,
hence
also
of
asser-
tion
(ii).
Now
assertion
(iii)
in
the
case
where
n
=
2
follows
immediately
from
assertion
(ii),
together
with
the
natural
inclusions
Out
FC
(Π
n
)
⊆
Out
FwC
(Π
n
)
⊆
Out
F
(Π
n
)
[cf.
Remark
2.1.1].
This
completes
the
proof
of
assertion
(iii).
Next,
we
verify
assertion
(i).
The
bijectivity
portion
of
assertion
(i)
follows
from
assertion
(ii),
together
with
the
bijectivity
portion
of
[NodNon],
Theorem
B.
Thus,
it
suffices
to
verify
the
injectivity
portion
of
assertion
(i).
First,
we
observe
that
injectivity
in
the
case
where
(g,
r)
=
(0,
3)
follows
from
assertion
(ii),
together
with
the
injectiv-
ity
portion
of
[NodNon],
Theorem
B.
Write
Π
†
2
(respectively,
Π
†
1
)
for
the
kernel
of
the
surjection
Π
n+1
Π
n−1
(respectively,
Π
n
Π
n−1
)
COMBINATORIAL
ANABELIAN
TOPICS
II
41
induced
by
the
projection
obtained
by
forgetting
the
factor(s)
labeled
def
n+1,
n
(respectively,
n).
Here,
if
n
=
1,
then
we
set
Π
n−1
=
Π
0
=
{1}.
Then
recall
[cf.
e.g.,
the
proof
of
[CmbCsp],
Theorem
4.1,
(i)]
that
we
have
natural
isomorphisms
out
out
Π
n+1
≃
Π
†
2
Π
n−1
;
Π
n
≃
Π
†
1
Π
n−1
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
Also,
we
recall
[cf.
[MzTa],
Proposition
2.4,
(i)]
that
one
may
interpret
the
surjection
Π
†
2
Π
†
1
induced
by
the
surjection
Π
n+1
Π
n
in
question
as
the
surjection
“Π
2
Π
1
”
that
arises
from
the
projection
X
2
→
X
in
the
case
of
an
“X”
of
type
(g,
r
+n−1).
Moreover,
one
verifies
easily
that
this
interpretation
is
compatible
with
the
definition
of
the
various
“Out(−)’s”
involved.
Thus,
since
n
inj
=
2
if
r
=
0,
the
above
natural
isomorphisms
allow
one
to
reduce
the
injectivity
in
question
to
the
case
where
n
=
1
and
r
=
0.
On
the
other
hand,
this
injectivity
follows
immediately
from
a
similar
argument
to
the
argument
used
in
the
proof
of
[CmbCsp],
Corollary
2.3,
(ii),
by
replacing
[CmbCsp],
Proposition
1.2,
(iii)
(respectively,
the
non-resp’d
portion
of
[CmbCsp],
Proposition
1.3,
(iv);
[CmbCsp],
Corollary
1.12,
(i)),
in
the
proof
of
[CmbCsp],
Corollary
2.3,
(ii),
by
Lemma
2.2,
(i)
(respectively,
Lemma
2.2,
(ii);
the
injectivity
in
question
in
the
case
where
(g,
r)
=
(0,
3),
which
was
verified
above).
This
completes
the
proof
of
the
injectivity
portion
of
assertion
(i),
hence
also
of
assertion
(i).
Finally,
assertion
(iv)
follows
immediately
from
assertion
(i),
to-
gether
with
a
similar
argument
to
the
argument
applied
in
the
proof
of
[CmbCsp],
Theorem
4.1,
(iv).
This
completes
the
proof
of
Theo-
rem
2.3.
Corollary
2.4
(PFC-admissibility
of
outomorphisms).
In
the
no-
tation
of
Theorem
2.3,
write
Out
PF
(Π
n
)
⊆
Out(Π
n
)
for
the
subgroup
of
PF-admissible
outomorphisms
[i.e.,
roughly
speak-
ing,
outomorphisms
that
preserve
the
fiber
subgroups
up
to
a
possible
permutation
of
the
factors
—
cf.
[CbTpI],
Definition
1.4,
(i)]
and
Out
PFC
(Π
n
)
⊆
Out
PF
(Π
n
)
for
the
subgroup
of
PFC-admissible
outomorphisms
[i.e.,
roughly
speak-
ing,
outomorphisms
that
preserve
the
fiber
subgroups
and
the
cuspidal
inertia
subgroups
up
to
a
possible
permutation
of
the
factors
—
cf.
[CbTpI],
Definition
1.4,
(iii)].
Let
us
regard
the
symmetric
group
on
n
letters
S
n
as
a
subgroup
of
Out(Π
n
)
via
the
natural
inclusion
of
The-
orem
2.3,
(iv).
Finally,
suppose
that
(g,
r)
∈
{(0,
3);
(1,
1)}.
Then
the
following
hold:
42
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(i)
We
have
an
equality
Out(Π
n
)
=
Out
PF
(Π
n
).
If,
moreover,
(r,
n)
=
(0,
2),
then
we
have
equalities
Out(Π
n
)
=
Out
PF
(Π
n
)
=
Out
F
(Π
n
)
×
S
n
[cf.
the
notational
conventions
introduced
in
Theorem
2.3].
(ii)
If
either
r
>
0,
n
≥
3
or
n
≥
4,
then
we
have
equalities
Out(Π
n
)
=
Out
PFC
(Π
n
)
=
Out
FC
(Π
n
)
×
S
n
[cf.
the
notational
conventions
introduced
in
Theorem
2.3].
Proof.
First,
we
verify
assertion
(i).
The
equality
in
the
first
display
of
assertion
(i)
follows
from
[MzTa],
Corollary
6.3,
together
with
the
assumption
that
(g,
r)
∈
{(0,
3);
(1,
1)}.
The
second
equality
in
the
second
display
of
assertion
(i)
follows
from
Theorem
2.3,
(iv).
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
The
first
equality
of
assertion
(ii)
follows
immediately
from
Theorem
2.3,
(ii),
together
with
the
first
equality
of
assertion
(i).
The
second
equality
of
assertion
(ii)
follows
from
[NodNon],
Theorem
B.
This
completes
the
proof
of
assertion
(ii).
Corollary
2.5
(Anabelian
properties
of
hyperbolic
curves
and
associated
configuration
spaces
I).
Let
Σ
be
a
set
of
prime
numbers
which
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardinality
one;
m
≤
n
positive
integers;
(g,
r)
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
k
a
field
of
characteristic
∈
Σ;
k
a
separable
closure
of
k;
X
a
hyperbolic
curve
of
type
(g,
r)
over
k.
Write
def
G
k
=
Gal(k/k).
For
each
positive
integer
i,
write
X
i
for
the
i-th
def
configuration
space
of
X;
(X
i
)
k
=
X
i
×
k
k;
Δ
X
i
for
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
of
(X
i
)
k
;
ρ
Σ
X
i
:
G
k
−→
Out(Δ
X
i
)
for
the
pro-Σ
outer
Galois
representation
associated
to
X
i
;
S
i
for
the
symmetric
group
on
i
letters;
Φ
i
:
S
i
−→
Out(Δ
X
i
)
for
the
outer
representation
arising
from
the
permutations
of
the
factors
of
X
i
.
Suppose
that
the
following
conditions
are
satisfied:
COMBINATORIAL
ANABELIAN
TOPICS
II
43
(1)
(g,
r)
∈
{(0,
3);
(1,
1)}.
(2)
If
(r,
n,
m)
∈
{(0,
2,
1);
(0,
2,
2);
(0,
3,
1)},
then
there
exists
an
l
∈
Σ
such
that
k
is
l-cyclotomically
full,
i.e.,
the
l-adic
cyclotomic
character
of
G
k
has
open
image.
Then
the
following
hold:
(i)
Let
α
∈
Out(Δ
X
n
).
Then
there
exists
a
unique
element
σ
α
∈
S
n
such
that
α
◦
Φ
n
(σ
α
)
∈
Out
F
(Δ
X
n
)
[cf.
the
notational
conventions
introduced
in
Theorem
2.3].
Write
α
m
∈
Out
F
(Δ
X
m
)
for
the
outomorphism
of
Δ
X
m
induced
by
α
◦
Φ
n
(σ
α
),
relative
to
the
quotient
Δ
X
n
Δ
X
m
by
a
fiber
subgroup
of
co-length
m
of
Δ
X
n
.
[Note
that
it
follows
from
[CbTpI],
Theorem
A,
(i),
that
α
m
does
not
depend
on
the
choice
of
fiber
subgroup
of
co-length
m
of
Δ
X
n
.]
(ii)
If
(r,
n,
m)
∈
{(0,
2,
1);
(0,
2,
2);
(0,
3,
1)},
then
PFC
C
Out(Δ
Xn
)
(Im(ρ
Σ
(Δ
X
n
)
X
n
))
⊆
Out
[cf.
the
notational
conventions
introduced
in
Corollary
2.4].
(iii)
The
map
Out(Δ
X
n
)
−→
Out(Δ
X
m
)
α
→
α
m
[cf.
(i)]
determines
an
exact
sequence
of
homomorphisms
of
profinite
groups
Φ
n
1
−→
S
n
−→
Out
PFC
(Δ
X
n
)
−→
Out(Δ
X
m
)
—
where
the
second
arrow
is
a
split
injection
whose
image
commutes
with
Out
FC
(Δ
X
n
)
and
has
trivial
intersection
with
Im(ρ
Σ
X
n
).
If
(r,
n)
=
(0,
2),
then
the
map
α
→
α
m
deter-
mines
a
sequence
of
homomorphisms
of
profinite
groups
Φ
n
Out(Δ
X
n
)
−→
Out(Δ
X
m
)
1
−→
S
n
−→
—
where
the
second
arrow
is
a
split
injection
whose
im-
age
commutes
with
Out
F
(Δ
X
n
)
and
has
trivial
intersec-
tion
with
Im(ρ
Σ
X
n
)
—
which
is
exact
if,
moreover,
(r,
n,
m)
=
(0,
3,
1).
(iv)
Let
α
∈
Out(Δ
X
n
).
If
(r,
n,
m)
∈
{(0,
2,
1);
(0,
3,
1)},
then
we
suppose
further
that
α
∈
Out
PFC
(Δ
X
n
),
which
is
the
case
if,
for
instance,
α
∈
C
Out(Δ
Xn
)
(Im(ρ
Σ
X
n
))
[cf.
(ii)].
Then
it
holds
that
α
∈
Z
Out(Δ
Xn
)
(Im(ρ
Σ
X
n
))
Σ
(respectively,
N
Out(Δ
Xn
)
(Im(ρ
Σ
X
n
))
;
C
Out(Δ
Xn
)
(Im(ρ
X
n
)))
44
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
if
and
only
if
α
m
∈
Z
Out(Δ
Xm
)
(Im(ρ
Σ
X
m
))
Σ
(respectively,
N
Out(Δ
Xm
)
(Im(ρ
Σ
X
m
))
;
C
Out(Δ
Xm
)
(Im(ρ
X
m
)))
.
(v)
For
each
positive
integer
i,
write
Aut
k
(X
i
)
for
the
group
of
au-
tomorphisms
of
X
i
over
k.
Then
if
the
natural
homomorphism
Aut
k
(X
m
)
−→
Z
Out(Δ
Xm
)
(Im(ρ
Σ
X
m
))
is
bijective,
then
the
natural
homomorphism
Aut
k
(X
n
)
−→
Z
Out(Δ
Xn
)
(Im(ρ
Σ
X
n
))
is
bijective.
(vi)
For
each
positive
integer
i,
write
Aut((X
i
)
k
/k)
for
the
group
of
automorphisms
of
(X
i
)
k
that
are
compatible
with
some
au-
tomorphism
of
k;
Aut
ρ
(G
k
)
for
the
group
of
automorphisms
of
G
k
that
preserve
Ker(ρ
Σ
X
1
)
⊆
G
k
[where
we
note
that,
by
[NodNon],
Corollary
6.2,
(i),
for
any
positive
integer
i,
it
holds
Σ
that
Ker(ρ
Σ
X
1
)
=
Ker(ρ
X
i
)].
Then
if
the
natural
homomorphism
Aut((X
m
)
k
/k)
−→
Aut
ρ
(G
k
)×
Aut(Im(ρ
Σ
X
))
N
Out(Δ
Xm
)
(Im(ρ
Σ
X
m
))
m
is
bijective,
then
the
natural
homomorphism
Aut((X
n
)
k
/k)
−→
Aut
ρ
(G
k
)
×
Aut(Im(ρ
Σ
X
))
N
Out(Δ
Xn
)
(Im(ρ
Σ
X
n
))
n
is
bijective.
Proof.
First,
we
verify
assertion
(i).
The
existence
of
such
a
σ
α
follows
from
the
fact
that
Out(Δ
X
n
)
=
Out
PF
(Δ
X
n
)
[cf.
Corollary
2.4,
(i),
to-
gether
with
assumption
(1)].
The
uniqueness
of
such
a
σ
α
follows
imme-
diately
from
the
easily
verified
faithfulness
of
the
action
of
S
n
,
via
Φ
n
,
on
the
set
of
fiber
subgroups
of
Δ
X
n
.
This
completes
the
proof
of
asser-
tion
(i).
Next,
we
verify
assertion
(ii).
Since
Out(Δ
X
n
)
=
Out
PF
(Δ
X
n
)
[cf.
Corollary
2.4,
(i),
together
with
assumption
(1)],
assertion
(ii)
follows
immediately
from
[CmbGC],
Corollary
2.7,
(i),
together
with
condition
(2).
This
completes
the
proof
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
First,
let
us
observe
that
it
fol-
lows
immediately
from
the
various
definitions
involved
that
Im(Φ
n
)
⊆
Out
PFC
(Δ
X
n
).
Now
since
Out(Δ
X
n
)
=
Out
PF
(Δ
X
n
)
[cf.
Corollary
2.4,
(i),
together
with
assumption
(1)],
and
Out
F
(Δ
X
n
)
is
normalized
by
Out
PF
(Δ
X
n
),
one
verifies
easily
[i.e.,
by
considering
the
action
of
ele-
ments
of
Out
PF
(Δ
X
n
)
on
the
set
of
fiber
subgroups
of
Δ
X
n
]
that
the
second
arrow
in
either
of
the
two
displayed
sequences
is
a
split
injec-
tion.
Moreover,
since
[as
is
easily
verified]
the
outer
action
of
G
k
,
via
ρ
Σ
X
n
,
on
Δ
X
n
fixes
every
fiber
subgroup
of
Δ
X
n
,
it
follows
immediately
from
the
faithfulness
of
the
action
of
S
n
,
via
Φ
n
,
on
the
set
of
fiber
subgroups
of
Δ
X
n
that
the
image
of
the
second
arrow
in
either
of
the
COMBINATORIAL
ANABELIAN
TOPICS
II
45
two
displayed
sequences
has
trivial
intersection
with
Im(ρ
Σ
X
n
).
Now
it
follows
from
[NodNon],
Theorem
B,
that
the
image
of
the
second
arrow
of
the
first
displayed
sequence
commutes
with
Out
FC
(Δ
X
n
);
in
partic-
ular,
one
verifies
easily
from
the
various
definitions
involved
[cf.
also
Corollary
2.4,
(i),
together
with
assumption
(1)]
that
the
third
arrow
of
the
first
displayed
sequence
is
a
homomorphism.
If
(r,
n)
=
(0,
2),
then
it
follows
from
Corollary
2.4,
(i),
together
with
assumption
(1),
that
the
image
of
the
second
arrow
of
the
second
displayed
sequence
commutes
with
Out
F
(Δ
X
n
);
in
particular,
one
verifies
easily
from
the
various
def-
initions
involved
[cf.
also
Corollary
2.4,
(i),
together
with
assumption
(1)]
that
the
third
arrow
of
the
second
displayed
sequence
is
a
homo-
morphism.
Now
if
(r,
m)
=
(0,
1),
then
it
follows
immediately
from
the
injectivity
portion
of
Theorem
2.3,
(i),
together
with
the
equality
Out(Δ
X
n
)
=
Out
PF
(Δ
X
n
)
[cf.
Corollary
2.4,
(i),
together
with
assump-
tion
(1)],
that
the
kernel
of
the
third
arrow
in
either
of
the
two
displayed
sequences
is
Im(Φ
n
).
Moreover,
if
(r,
n,
m)
∈
{(0,
2,
1);
(0,
3,
1)},
then
it
follows
immediately
from
the
injectivity
portion
of
[NodNon],
Theo-
rem
B,
that
the
kernel
of
the
third
arrow
in
the
first
displayed
sequence
is
Im(Φ
n
).
On
the
other
hand,
if
(r,
m)
=
(0,
1)
and
n
∈
{2,
3},
then
it
follows
immediately
from
the
injectivity
portion
of
[NodNon],
Theorem
B,
together
with
Corollary
2.4,
(ii),
together
with
assumption
(1),
that
the
kernel
of
the
third
arrow
in
either
of
the
two
displayed
sequences
is
Im(Φ
n
).
This
completes
the
proof
of
assertion
(iii).
Next,
we
verify
assertion
(iv).
Now
since
the
permutations
of
the
factors
of
X
n
give
rise
to
automorphisms
of
X
n
over
k,
it
follows
im-
mediately
that
Im(Φ
n
)
⊆
Z
Out(Δ
Xn
)
(Im(ρ
Σ
X
n
)).
In
particular,
to
verify
assertion
(iv),
we
may
assume
without
loss
of
generality
—
by
replacing
α
by
α
n
[cf.
assertion
(i)]
—
that
α
∈
Out
F
(Δ
X
n
),
and
that
m
<
n.
Then
necessity
follows
immediately.
On
the
other
hand,
sufficiency
follows
immediately
from
the
exact
sequences
of
assertion
(iii).
This
completes
the
proof
of
assertion
(iv).
Assertion
(v)
(respectively,
(vi))
follows
immediately
from
assertions
(i),
(ii),
(iii),
(iv),
together
with
Lemma
2.7,
(iii),
below
(respectively,
Lemma
2.7,
(iv),
below).
This
completes
the
proof
of
Corollary
2.5.
Corollary
2.6
(Anabelian
properties
of
hyperbolic
curves
and
associated
configuration
spaces
II).
Let
Σ
be
a
set
of
prime
num-
bers
which
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardi-
nality
one;
m
≤
n
positive
integers;
(g
X
,
r
X
),
(g
Y
,
r
Y
)
pairs
of
non-
negative
integers
such
that
2g
X
−
2
+
r
X
,
2g
Y
−
2
+
r
Y
>
0;
k
X
,
k
Y
fields;
k
X
,
k
Y
separable
closures
of
k
X
,
k
Y
,
respectively;
X,
Y
hy-
perbolic
curves
of
type
(g
X
,
r
X
),
(g
Y
,
r
Y
)
over
k
X
,
k
Y
,
respectively.
def
def
Write
G
k
X
=
Gal(k
X
/k
X
);
G
k
Y
=
Gal(k
Y
/k
Y
).
For
each
positive
integer
i,
write
X
i
,
Y
i
for
the
i-th
configuration
spaces
of
X,
Y
,
46
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
def
def
respectively;
(X
i
)
k
X
=
X
i
×
k
X
k
X
;
(Y
i
)
k
Y
=
Y
i
×
k
Y
k
Y
;
π
1
Σ
((X
i
)
k
X
),
π
1
Σ
((Y
i
)
k
Y
)
for
the
maximal
pro-Σ
quotients
of
the
étale
fundamental
(Σ)
groups
π
1
((X
i
)
k
X
),
π
1
((Y
i
)
k
Y
)
of
(X
i
)
k
X
,
(Y
i
)
k
Y
,
respectively;
π
1
(X
i
),
(Σ)
π
1
(Y
i
)
for
the
geometrically
pro-Σ
étale
fundamental
groups
of
X
i
,
Y
i
,
respectively,
i.e.,
the
quotients
of
the
étale
fundamental
groups
π
1
(X
i
),
π
1
(Y
i
)
of
X
i
,
Y
i
by
the
respective
kernels
of
the
natural
surjections
π
1
((X
i
)
k
X
)
π
1
Σ
((X
i
)
k
X
),
π
1
((Y
i
)
k
Y
)
π
1
Σ
((Y
i
)
k
Y
).
Suppose
that
the
following
conditions
are
satisfied:
(1)
{(g
X
,
r
X
);
(g
Y
,
r
Y
)}
∩
{(0,
3);
(1,
1)}
=
∅.
(2)
If
(r
X
,
n,
m)
(respectively,
(r
Y
,
n,
m))
is
contained
in
the
set
{(0,
2,
1);
(0,
2,
2);
(0,
3,
1)},
then
there
exists
an
l
∈
Σ
such
that
k
X
(respectively,
k
Y
)
is
l-cyclotomically
full,
i.e.,
the
l-
adic
cyclotomic
character
of
G
k
X
(respectively,
G
k
Y
)
has
open
image.
Then
the
following
hold:
∼
(i)
Let
θ
:
k
X
→
k
Y
be
an
isomorphism
of
fields
that
determines
∼
an
isomorphism
k
X
→
k
Y
.
For
each
positive
integer
i,
write
Isom
θ
(X
i
,
Y
i
)
for
the
set
of
isomorphisms
of
X
i
with
Y
i
that
∼
are
compatible
with
the
isomorphism
k
X
→
k
Y
determined
(Σ)
(Σ)
by
θ;
Isom
θ
(π
1
(X
i
),
π
1
(Y
i
))
for
the
set
of
isomorphisms
of
(Σ)
(Σ)
π
1
(X
i
)
with
π
1
(Y
i
)
that
are
compatible
with
the
isomorphism
∼
G
k
X
→
G
k
Y
determined
by
θ.
Then
if
the
natural
map
(Σ)
(Σ)
Isom
θ
(X
m
,
Y
m
)
−→
Isom
θ
(π
1
(X
m
),
π
1
(Y
m
))/Inn(π
1
Σ
((Y
m
)
k
Y
))
is
bijective,
then
the
natural
map
(Σ)
(Σ)
Isom
θ
(X
n
,
Y
n
)
−→
Isom
θ
(π
1
(X
n
),
π
1
(Y
n
))/Inn(π
1
Σ
((Y
n
)
k
Y
))
is
bijective.
(ii)
For
each
positive
integer
i,
write
Isom((X
i
)
k
X
/k
X
,
(Y
i
)
k
Y
/k
Y
)
for
the
set
of
isomorphisms
of
(X
i
)
k
X
with
(Y
i
)
k
Y
that
are
com-
patible
with
some
field
isomorphism
of
k
X
with
k
Y
;
(Σ)
(Σ)
Isom(π
1
(X
i
)/G
k
X
,
π
1
(Y
i
)/G
k
Y
)
(Σ)
(Σ)
for
the
set
of
isomorphisms
of
π
1
(X
i
)
with
π
1
(Y
i
)
that
are
compatible
with
some
isomorphism
of
G
k
X
with
G
k
Y
.
Then
if
the
natural
map
Isom((X
m
)
k
X
/k
X
,
(Y
m
)
k
Y
/k
Y
)
(Σ)
(Σ)
−→
Isom(π
1
(X
m
)/G
k
X
,
π
1
(Y
m
)/G
k
Y
)/Inn(π
1
Σ
((Y
m
)
k
Y
))
is
bijective,
then
the
natural
map
Isom((X
n
)
k
X
/k
X
,
(Y
n
)
k
Y
/k
Y
)
COMBINATORIAL
ANABELIAN
TOPICS
II
(Σ)
47
(Σ)
−→
Isom(π
1
(X
n
)/G
k
X
,
π
1
(Y
n
)/G
k
Y
)/Inn(π
1
Σ
((Y
n
)
k
Y
))
is
bijective.
Proof.
Consider
assertion
(i)
(respectively,
(ii)).
If
the
set
(Σ)
(Σ)
Isom
θ
(π
1
(X
n
),
π
1
(Y
n
))/Inn(π
1
Σ
((Y
n
)
k
Y
))
(respectively,
(Σ)
(Σ)
Isom(π
1
(X
n
)/G
k
X
,
π
1
(Y
n
)/G
k
Y
)/Inn(π
1
Σ
((Y
n
)
k
Y
))
)
is
empty,
then
assertion
(i)
(respectively,
(ii))
is
immediate.
Thus,
we
may
suppose
without
loss
of
generality
that
this
set
is
nonempty.
Then
one
verifies
easily
from
[MzTa],
Corollary
6.3,
together
with
condition
(1),
that
the
set
(Σ)
(Σ)
Isom
θ
(π
1
(X
m
),
π
1
(Y
m
))/Inn(π
1
Σ
((Y
m
)
k
Y
))
(respectively,
(Σ)
(Σ)
Isom(π
1
(X
m
)/G
k
X
,
π
1
(Y
m
)/G
k
Y
)/Inn(π
1
Σ
((Y
m
)
k
Y
))
)
is
nonempty.
Thus,
it
follows
immediately
from
the
bijectivity
assumed
in
assertion
(i)
(respectively,
(ii))
that
there
exists
an
isomorphism
∼
∼
X
m
→
Y
m
that
is
compatible
with
the
isomorphism
k
X
→
k
Y
deter-
∼
mined
by
θ
(respectively,
an
isomorphism
(X
m
)
k
X
→
(Y
m
)
k
Y
that
is
∼
compatible
with
some
isomorphism
k
X
→
k
Y
).
In
particular,
it
follows
immediately
from
Lemma
2.7,
(iii),
below
(respectively,
Lemma
2.7,
∼
(iv),
below)
that
there
exists
an
isomorphism
X
→
Y
that
is
compatible
∼
with
the
isomorphism
k
X
→
k
Y
determined
by
θ
(respectively,
an
iso-
∼
morphism
X
×
k
X
k
X
→
Y
×
k
Y
k
Y
that
is
compatible
with
some
isomor-
∼
phism
k
X
→
k
Y
).
Thus,
by
pulling
back
the
various
objects
involved
via
this
isomorphism,
to
verify
assertion
(i)
(respectively,
(ii)),
we
may
assume
without
loss
of
generality
that
(X,
k
X
,
k
X
,
θ)
=
(Y,
k
Y
,
k
Y
,
id
k
X
)
(respectively,
(X,
k
X
,
k
X
)
=
(Y,
k
Y
,
k
Y
)).
Then
assertion
(i)
(respec-
tively,
(ii))
follows
from
Corollary
2.5,
(v)
(respectively,
Corollary
2.5,
(vi)).
This
completes
the
proof
of
Corollary
2.6.
Lemma
2.7
(Isomorphisms
between
configuration
spaces
of
hyperbolic
curves).
Let
n
be
a
positive
integer;
(g
X
,
r
X
),
(g
Y
,
r
Y
)
pairs
of
nonnegative
integers
such
that
2g
X
−
2
+
r
X
,
2g
Y
−
2
+
r
Y
>
0;
k
X
,
k
Y
fields;
k
X
,
k
Y
separable
closures
of
k
X
,
k
Y
,
respectively;
X,
Y
hyperbolic
curves
of
type
(g
X
,
r
X
),
(g
Y
,
r
Y
)
over
k
X
,
k
Y
,
respectively.
Write
X
n
,
Y
n
for
the
n-th
configuration
spaces
of
X,
Y
,
respectively;
def
def
def
def
X
k
X
=
X
×
k
X
k
X
;
Y
k
Y
=
Y
×
k
Y
k
Y
;
(X
n
)
k
X
=
X
n
×
k
X
k
X
;
(Y
n
)
k
Y
=
48
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Y
n
×
k
Y
k
Y
;
S
n
for
the
symmetric
group
on
n
letters;
Aut
k
X
(X
n
)
for
the
group
of
automorphisms
of
X
n
over
k
X
;
Ψ
n
:
S
n
−→
Aut
k
X
(X
n
)
for
the
action
of
S
n
on
X
n
over
k
X
obtained
by
permuting
the
factors
of
X
n
.
Suppose
that
(g
X
,
r
X
),
(g
Y
,
r
Y
)
∈
{(0,
3);
(1,
1)}.
Then
the
following
hold:
∼
(i)
Let
α
:
X
n
→
Y
n
be
an
isomorphism.
Then
there
exists
a
∼
unique
isomorphism
α
0
:
k
Y
→
k
X
that
is
compatible
with
α
relative
to
the
structure
morphisms
of
X
n
,
Y
n
.
∼
(ii)
Let
α
:
X
n
→
Y
n
be
an
isomorphism.
Then
there
exist
a
unique
permutation
σ
∈
Ψ
n
(S
n
)
⊆
Aut
k
X
(X
n
)
and
a
unique
isomor-
∼
phism
α
1
:
X
→
Y
that
is
compatible
with
α
◦
σ
relative
to
the
projections
X
n
→
X,
Y
n
→
Y
to
each
of
the
n
factors.
(iii)
Write
Isom(X
n
,
Y
n
)
for
the
set
of
isomorphisms
of
X
n
with
Y
n
;
def
Isom(X,
Y
)
=
Isom(X
1
,
Y
1
).
Then
the
natural
map
Isom(X,
Y
)
×
Ψ
n
(S
n
)
−→
Isom(X
n
,
Y
n
)
is
bijective.
(iv)
Write
Isom((X
n
)
k
X
/k
X
,
(Y
n
)
k
Y
/k
Y
)
for
the
set
of
isomorphisms
∼
(X
n
)
k
X
→
(Y
n
)
k
Y
that
are
compatible
with
some
isomorphism
∼
def
k
Y
→
k
X
;
Isom(X
k
X
/k
X
,
Y
k
Y
/k
Y
)
=
Isom((X
1
)
k
X
/k
X
,
(Y
1
)
k
Y
/k
Y
).
Then
the
natural
map
Isom(X
k
X
/k
X
,
Y
k
Y
/k
Y
)
×
Ψ
n
(S
n
)
−→
Isom((X
n
)
k
X
/k
X
,
(Y
n
)
k
Y
/k
Y
)
is
bijective.
Proof.
First,
we
verify
assertion
(i).
Write
(C
n
X
)
log
,
(C
n
Y
)
log
for
the
n-th
log
configuration
spaces
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”]
of
[the
smooth
log
curves
over
k
X
,
k
Y
determined
by]
X,
Y
,
respectively.
Then
recall
[cf.
the
discussion
at
the
beginning
of
[MzTa],
§2]
that
(C
n
X
)
log
,
(C
n
Y
)
log
are
log
regular
log
schemes
whose
interiors
are
naturally
isomorphic
to
X
n
,
Y
n
,
respec-
tively,
and
that
the
underlying
schemes
C
n
X
,
C
n
Y
of
(C
n
X
)
log
,
(C
n
Y
)
log
are
proper
over
k
X
,
k
Y
,
respectively.
Thus,
by
applying
[ExtFam],
Theorem
A,
(1),
to
the
composite
α
X
n
→
Y
n
→
C
n
Y
→
M
g
Y
,r
Y
+n
—
where
we
refer
to
the
discussion
entitled
“Curves”
in
[CbTpI],
§0,
concerning
the
notation
“M
g
Y
,r
Y
+n
”;
the
third
arrow
is
the
natural
(1-)morphism
arising
from
the
definition
of
C
n
Y
—
we
conclude
that
the
composite
α
X
n
→
Y
n
→
C
n
Y
→
M
g
Y
,r
Y
+n
→
(M
g
Y
,r
Y
+n
)
c
COMBINATORIAL
ANABELIAN
TOPICS
II
49
—
where
we
write
(M
g
Y
,r
Y
+n
)
c
for
the
coarse
moduli
space
associated
to
M
g
Y
,r
Y
+n
—
factors
through
the
natural
open
immersion
X
n
→
C
n
X
.
On
the
other
hand,
one
verifies
immediately
that
the
composite
C
n
Y
→
M
g
Y
,r
Y
+n
→
(M
g
Y
,r
Y
+n
)
c
is
proper
and
quasi-finite,
hence
finite.
In
particular,
if
we
write
C
Γ
⊆
C
n
X
×
k
C
n
Y
for
the
scheme-theoretic
clo-
α
sure
of
the
graph
of
the
composite
X
n
→
Y
n
→
C
n
Y
,
then
the
composite
pr
C
Γ
→
C
n
X
×
k
C
n
Y
→
1
C
n
X
is
a
finite
morphism
from
an
integral
scheme
to
a
normal
scheme
which
induces
an
isomorphism
between
the
respective
function
fields.
Thus,
we
conclude
that
this
composite
is
an
isomor-
phism,
hence
that
α
extends
uniquely
to
a
morphism
C
n
X
→
C
n
Y
.
Now
recall
that
C
n
X
is
proper,
geometrically
normal,
and
geometrically
con-
nected
over
k
X
.
Thus,
one
verifies
immediately,
by
considering
global
sections
of
the
respective
structure
sheaves,
that
there
exists
a
unique
homomorphism
α
0
:
k
Y
→
k
X
that
is
compatible
with
α.
Moreover,
by
applying
a
similar
argument
to
α
−1
,
it
follows
that
α
0
is
an
isomor-
phism.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
First,
let
us
observe
that,
by
replacing
∼
Y
by
the
result
of
base-changing
Y
via
α
0
:
k
Y
→
k
X
[cf.
assertion
(i)],
we
may
assume
without
loss
of
generality
that
k
Y
=
k
X
,
k
Y
=
k
X
,
and
that
α
is
an
isomorphism
over
k
X
.
Next,
let
us
observe
that
it
is
immediate
that
σ
and
α
1
as
in
the
statement
of
assertion
(ii)
are
unique;
thus,
it
remains
to
verify
the
existence
of
such
σ
and
α
1
.
Next,
let
us
observe
that
it
follows
immediately
from
[MzTa],
Corollary
6.3,
that
there
exists
a
permutation
σ
∈
Ψ
n
(S
n
)
such
that
if
we
identify
the
respective
sets
of
fiber
subgroups
of
Δ
X
n
,
Δ
Y
n
—
where
we
write
Δ
X
n
,
Δ
Y
n
for
the
maximal
pro-l
quotients
of
the
étale
fundamental
groups
of
(X
n
)
k
X
,
(Y
n
)
k
X
,
respectively,
for
some
prime
number
l
that
is
invertible
in
k
X
—
with
the
set
2
{1,···
,n}
[cf.
the
discussion
entitled
“Sets”
in
[CbTpI],
§0]
in
the
evident
way,
then
the
automorphism
of
def
the
set
2
{1,···
,n}
induced
by
the
composite
β
=
α
◦
σ
is
the
identity
automorphism.
Write
pr
X
:
X
n
→
X,
pr
Y
:
Y
n
→
Y
for
the
projections
to
the
factor
labeled
n,
respectively.
Then
we
claim
that
the
following
assertion
holds:
∼
Claim
2.7.A:
There
exists
an
isomorphism
α
1
:
X
→
Y
that
is
compatible
with
β
relative
to
pr
X
,
pr
Y
.
Indeed,
write
Γ
⊆
X
×
k
X
Y
for
the
scheme-theoretic
image
via
X
n
×
k
X
(pr
,id
Y
)
β
pr
X
Y
−→
X
×
k
X
Y
of
the
graph
of
the
composite
X
n
→
Y
n
→
Y
Y
.
Next,
let
us
observe
that
if
Z
is
an
irreducible
scheme
of
finite
type
over
k
X
,
then
any
nonconstant
[i.e.,
dominant]
k
X
-morphism
Z
→
Y
k
X
induces
an
open
homomorphism
between
the
respective
fundamental
groups.
Thus,
since
the
automorphism
of
the
set
2
{1,···
,n}
induced
by
β
is
the
identity
automorphism,
it
follows
immediately
that,
for
any
k
X
-valued
geometric
point
x
of
X,
if
we
write
F
for
the
geometric
50
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
β
k
fiber
of
pr
X
:
X
n
→
X
at
x,
then
the
composite
F
→
(X
n
)
k
X
→
X
(pr
Y
)
k
(Y
n
)
k
X
→
X
Y
k
X
is
constant.
In
particular,
one
verifies
immediately
that
Γ
is
an
integral,
separated
scheme
of
dimension
1.
Thus,
since
pr
X
is
surjective,
geometrically
connected,
smooth,
and
factors
through
pr
the
composite
Γ
→
X
×
k
X
Y
→
1
X,
it
follows
immediately
that
this
composite
morphism
Γ
→
X
is
surjective
and
induces
an
isomorphism
between
the
respective
function
fields.
Therefore,
one
concludes
easily,
by
applying
Zariski’s
main
theorem,
that
the
composite
Γ
→
X
×
k
X
pr
Y
→
1
X
is
an
isomorphism,
hence
that
there
exists
a
unique
morphism
α
1
:
X
→
Y
such
that
pr
Y
◦
β
=
α
1
◦
pr
X
.
Moreover,
by
applying
a
similar
argument
to
β
−1
,
it
follows
that
α
1
is
an
isomorphism.
This
completes
the
proof
of
Claim
2.7.A.
∼
Write
γ
for
the
composite
of
β
with
the
isomorphism
Y
n
→
X
n
de-
termined
by
α
1
−1
.
Then
it
is
immediate
that
γ
is
an
automorphism
of
X
n
over
X
relative
to
pr
X
;
in
particular,
the
outomorphism
of
Δ
X
n
induced
by
γ
is
contained
in
the
kernel
of
the
homomorphism
Out
F
(Δ
X
n
)
→
Out
F
(Δ
X
)
—
where
we
write
Δ
X
for
the
maximal
pro-l
quotient
of
the
étale
fundamental
group
of
X
k
X
—
induced
by
pr
X
.
Now,
by
applying
a
similar
argument
to
the
argument
of
the
proof
of
Claim
2.7.A,
one
verifies
easily
that,
for
each
i
∈
{1,
·
·
·
,
n},
there
ex-
ists
an
automorphism
γ
1,i
of
X
that
is
compatible
with
γ
relative
to
the
projection
X
n
→
X
to
the
factor
labeled
i.
[Thus,
γ
1,n
=
id
X
.]
More-
over,
since,
by
applying
induction
on
n,
we
may
assume
that
assertion
(ii)
has
already
been
verified
for
n
−
1,
it
follows
immediately
that
the
outomorphism
of
Δ
X
n
induced
by
γ
is
contained
in
Out
FC
(Δ
X
n
),
hence
in
the
kernel
of
the
homomorphism
Out
FC
(Δ
X
n
)
→
Out
FC
(Δ
X
)
induced
by
the
projections
X
n
→
X
to
each
of
the
n
factors
[cf.
[CmbCsp],
Proposition
1.2,
(iii)].
Therefore,
it
follows
immediately
from
the
argument
of
the
first
paragraph
of
the
proof
of
[LocAn],
The-
orem
14.1,
that,
for
each
i
∈
{1,
·
·
·
,
n},
γ
1,i
is
the
identity
automor-
phism
of
X,
hence
also
that
γ
is
the
identity
automorphism
of
X
n
.
This
completes
the
proof
of
assertion
(ii).
Assertions
(iii),
(iv)
follow
immediately
from
assertion
(ii),
together
with
the
various
definitions
involved.
This
completes
the
proof
of
Lemma
2.7.
COMBINATORIAL
ANABELIAN
TOPICS
II
51
3.
Synchronization
of
tripods
In
the
present
§3,
we
introduce
and
study
the
notion
of
a
tripod
of
the
log
fundamental
group
of
the
log
configuration
space
of
a
stable
log
curve
[cf.
Definition
3.3,
(i),
below].
In
particular,
we
discuss
the
phenomenon
of
synchronization
among
the
various
tripods
of
the
log
fundamental
group
[cf.
Theorems
3.17;
3.18,
below].
One
interesting
consequence
of
this
phenomenon
of
tripod
synchronization
is
a
certain
non-surjectivity
result
[cf.
Corollary
3.22
below].
Finally,
we
apply
the
theory
of
synchronization
of
tripods
to
show
that,
under
certain
conditions,
commuting
profinite
Dehn
multi-twists
are
“co-Dehn”
[cf.
Corollary
3.25
below]
and
to
compute
the
commensurator
of
certain
purely
combinatorial/group-theoretic
groups
of
profinite
Dehn
multi-
twists
in
terms
of
scheme
theory
[cf.
Corollary
3.27
below].
In
the
present
§3,
let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
n
a
positive
integer;
Σ
a
set
of
prime
numbers
which
is
either
the
set
of
all
prime
numbers
or
of
cardinality
one;
k
an
algebraically
closed
field
of
characteristic
∈
Σ;
(Spec
k)
log
the
log
scheme
obtained
by
equipping
Spec
k
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
=
X
1
log
a
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
.
For
each
[possibly
empty]
subset
E
⊆
{1,
·
·
·
,
n},
write
X
E
log
for
the
#E-th
log
configuration
space
of
the
stable
log
curve
X
log
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”],
where
we
think
of
the
factors
as
being
labeled
by
the
elements
of
E
⊆
{1,
·
·
·
,
n};
Π
E
for
the
maximal
pro-Σ
quotient
of
the
kernel
of
the
natural
surjection
π
1
(X
E
log
)
π
1
((Spec
k)
log
).
Thus,
by
applying
a
suitable
specializa-
tion
isomorphism
—
cf.
the
discussion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1
—
one
verifies
easily
that
Π
E
is
equipped
with
a
natural
structure
of
pro-Σ
configuration
space
group
—
cf.
[MzTa],
Definition
2.3,
(i).
For
each
1
≤
m
≤
n,
write
def
def
log
log
=
X
{1,···
X
m
,m}
;
Π
m
=
Π
{1,···
,m}
.
Thus,
for
subsets
E
⊆
E
⊆
{1,
·
·
·
,
n},
we
have
a
projection
log
log
p
log
E/E
:
X
E
→
X
E
obtained
by
forgetting
the
factors
that
belong
to
E
\
E
.
For
E
⊆
E
⊆
{1,
·
·
·
,
n}
and
1
≤
m
≤
m
≤
n,
we
shall
write
p
Π
E/E
:
Π
E
Π
E
52
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
for
some
fixed
surjection
[that
belongs
to
the
collection
of
surjections
that
constitutes
the
outer
surjection]
induced
by
p
log
E/E
;
def
Π
E/E
=
Ker(p
Π
E/E
)
⊆
Π
E
;
def
log
log
log
p
log
m/m
=
p
{1,···
,m}/{1,···
,m
}
:
X
m
−→
X
m
;
def
Π
p
Π
m/m
=
p
{1,···
,m}/{1,···
,m
}
:
Π
m
Π
m
;
def
Π
m/m
=
Π
{1,···
,m}/{1,···
,m
}
⊆
Π
m
.
Finally,
recall
[cf.
the
statement
of
Theorem
2.3,
(iv)]
the
natural
inclusion
S
n
→
Out(Π
n
)
—
where
we
write
S
n
for
the
symmetric
group
on
n
letters
—
obtained
by
permuting
the
various
factors
of
X
n
.
We
shall
regard
S
n
as
a
subgroup
of
Out(Π
n
)
by
means
of
this
natural
inclusion.
Definition
3.1.
Let
i
∈
E
⊆
{1,
·
·
·
,
n};
x
∈
X
n
(k)
a
k-valued
geo-
metric
point
of
the
underlying
scheme
X
n
of
X
n
log
.
(i)
Let
E
⊆
{1,
·
·
·
,
n}
be
a
subset.
Then
we
shall
write
x
E
∈
X
E
(k)
for
the
k-valued
geometric
point
of
X
E
obtained
by
forming
the
image
of
x
∈
X
n
(k)
via
p
{1,···
,n}/E
:
X
n
→
X
E
;
def
log
x
log
E
=
x
E
×
X
E
X
E
.
(ii)
We
shall
write
G
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
deter-
mined
by
the
stable
log
curve
X
log
over
(Spec
k)
log
[cf.
[CmbGC],
Example
2.5];
G
for
the
underlying
semi-graph
of
G;
Π
G
for
the
[pro-Σ]
fundamental
group
of
G;
G
−→
G
for
the
universal
covering
of
G
corresponding
to
Π
G
.
Thus,
we
have
a
natural
outer
isomorphism
∼
Π
1
−→
Π
G
.
Throughout
our
discussion
of
the
objects
introduced
at
the
∼
beginning
of
the
present
§3,
let
us
fix
an
isomorphism
Π
1
→
Π
G
that
belongs
to
the
collection
of
isomorphisms
that
constitutes
the
above
natural
outer
isomorphism.
COMBINATORIAL
ANABELIAN
TOPICS
II
53
(iii)
We
shall
write
G
i∈E,x
for
the
semi-graph
of
anabelioids
of
pro-Σ
PSC-type
deter-
log
mined
by
the
geometric
fiber
of
the
projection
p
log
E/(E\{i})
:
X
E
→
log
log
over
x
log
X
E\{i}
E\{i}
→
X
E\{i}
[cf.
(i)];
Π
G
i∈E,x
for
the
[pro-Σ]
fundamental
group
of
G
i∈E,x
.
Thus,
we
have
a
natural
identification
G
=
G
i∈{i},x
and
a
natural
Π
E
-orbit
[i.e.,
relative
to
composition
with
au-
tomorphisms
induced
by
conjugation
by
elements
of
Π
E
]
of
isomorphisms
∼
(Π
E
⊇)
Π
E/(E\{i})
−→
Π
G
i∈E,x
.
Throughout
our
discussion
of
the
objects
introduced
at
the
beginning
of
the
present
§3,
let
us
fix
an
outer
isomorphism
∼
Π
E/(E\{i})
−→
Π
G
i∈E,x
whose
constituent
isomorphisms
belong
to
the
Π
E
-orbit
of
iso-
morphisms
just
discussed.
(iv)
Let
v
∈
Vert(G
i∈E,x
)
(respectively,
e
∈
Cusp(G
i∈E,x
);
e
∈
Node(G
i∈E,x
);
e
∈
Edge(G
i∈E,x
);
z
∈
VCN(G
i∈E,x
)).
Then
we
shall
refer
to
the
image
[in
Π
E
]
of
a
verticial
(respectively,
a
cuspidal;
a
nodal;
an
edge-like;
a
VCN-)
subgroup
[cf.
[CbTpI],
Definition
2.1,
(i)]
of
Π
G
i∈E,x
associated
to
v
(respectively,
e;
e;
∼
e;
z)
via
the
inverse
Π
G
i∈E,x
→
Π
E/(E\{i})
⊆
Π
E
of
any
isomor-
phism
that
lifts
the
fixed
outer
isomorphism
discussed
in
(iii)
as
a
verticial
(respectively,
a
cuspidal;
a
nodal;
an
edge-like;
a
VCN-)
subgroup
of
Π
E
associated
to
v
(respectively,
e;
e;
e;
z).
Thus,
the
notion
of
a
verticial
(respectively,
a
cuspidal;
a
nodal;
an
edge-like;
a
VCN-)
subgroup
of
Π
E
associated
to
v
(respectively,
e;
e;
e;
z)
depends
on
the
choice
of
the
fixed
outer
isomorphism
of
(iii)
[but
cf.
Lemma
3.2,
(i),
below,
in
the
case
of
cusps!].
(v)
We
shall
say
that
a
vertex
v
∈
Vert(G
i∈E,x
)
of
G
i∈E,x
is
a(n)
[E-]tripod
of
X
n
log
if
v
is
of
type
(0,
3)
[cf.
[CbTpI],
Definition
2.3,
(iii)].
If,
in
this
situation,
C(v)
=
∅,
then
we
shall
say
that
the
tripod
v
is
cusp-supporting.
(vi)
We
shall
say
that
a
cusp
c
∈
Cusp(G
i∈E,x
)
of
G
i∈E,x
is
diagonal
if
c
does
not
arise
from
a
cusp
of
the
copy
of
X
log
given
by
the
factor
of
X
E
log
labeled
i
∈
E.
54
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Lemma
3.2
(Cusps
of
various
fibers).
Let
i
∈
E
⊆
{1,
·
·
·
,
n};
x
∈
X
n
(k).
Then
the
following
hold:
∼
(i)
Let
c
∈
Cusp(G
i∈E,x
)
and
Π
c
⊆
Π
G
i∈E,x
←
Π
E/(E\{i})
a
cuspidal
∼
subgroup
of
Π
G
i∈E,x
←
Π
E/(E\{i})
associated
to
c
∈
Cusp(G
i∈E,x
).
Then
any
Π
E
-conjugate
of
Π
c
is,
in
fact,
a
Π
E/(E\{i})
-conju-
gate
of
Π
c
.
(ii)
Each
diagonal
cusp
of
G
i∈E,x
[cf.
Definition
3.1,
(vi)]
admits
a
natural
label
∈
E
\
{i}.
More
precisely,
for
each
j
∈
E
\
{i},
there
exists
a
unique
diagonal
cusp
of
G
i∈E,x
that
arises
from
the
divisor
of
the
fiber
product
over
k
of
#E
copies
of
X
consisting
of
the
points
whose
i-th
and
j-th
factors
coincide.
(iii)
Let
α
∈
Aut
F
(Π
n
)
[cf.
[CmbCsp],
Definition
1.1,
(ii)].
Sup-
pose
that
either
E
=
{1,
·
·
·
,
n}
or
n
≥
n
FC
[cf.
Theorem
2.3,
∼
(ii)].
Then
the
outomorphism
of
Π
G
i∈E,x
←
Π
E/(E\{i})
deter-
mined
by
α
is
group-theoretically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(iv)].
(iv)
Let
α
∈
Aut
F
(Π
n
)
and
c
∈
Cusp(G
i∈E,x
)
a
diagonal
cusp
of
∼
G
i∈E,x
.
Suppose
that
the
outomorphism
of
Π
G
i∈E,x
←
Π
E/(E\{i})
determined
by
α
is
group-theoretically
cuspidal.
Then
this
outomorphism
preserves
the
Π
G
i∈E,x
-conjugacy
class
of
∼
cuspidal
subgroups
of
Π
G
i∈E,x
←
Π
E/(E\{i})
associated
to
c
∈
Cusp(G
i∈E,x
).
Proof.
Assertion
(i)
follows
immediately
from
the
[easily
verified]
fact
that
the
restriction
of
p
Π
E/(E\{i})
:
Π
E
Π
E\{i}
to
the
normalizer
of
Π
c
in
Π
E
is
surjective.
Assertion
(ii)
follows
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(iii).
If
E
=
{1,
·
·
·
,
n}
(respectively,
n
≥
n
FC
),
then
assertion
(iii)
follows
immediately
from
[CbTpI],
Theorem
A,
(ii)
(respectively,
Theorem
2.3,
(ii),
of
the
present
monograph),
together
with
assertion
(i).
This
completes
the
proof
of
assertion
(iii).
Finally,
assertion
(iv)
follows
immediately
from
the
definition
of
F-admissibility
[cf.
also
assertion
(ii)].
This
completes
the
proof
of
Lemma
3.2.
Definition
3.3.
Let
E
⊆
{1,
·
·
·
,
n}.
(i)
We
shall
say
that
a
closed
subgroup
H
⊆
Π
E
of
Π
E
is
a(n)
[E-]tripod
of
Π
n
if
H
is
a
verticial
subgroup
of
Π
E
[cf.
Def-
inition
3.1,
(iv)]
associated
to
a(n)
[E-]tripod
v
of
X
n
log
[cf.
Definition
3.1,
(v)].
If,
in
this
situation,
the
tripod
v
is
cusp-
supporting
[cf.
Definition
3.1,
(v)],
then
we
shall
say
that
the
tripod
H
is
cusp-supporting.
COMBINATORIAL
ANABELIAN
TOPICS
II
55
(ii)
We
shall
say
that
an
E-tripod
of
Π
n
[cf.
(i)]
is
trigonal
if,
for
every
j
∈
E,
the
image
of
the
tripod
via
p
Π
E/{j}
:
Π
E
Π
{j}
is
trivial.
(iii)
Let
T
⊆
Π
E
be
an
E-tripod
of
Π
n
[cf.
(i)]
and
E
⊆
E.
Then
we
shall
say
that
T
is
E
-strict
if
the
image
p
Π
E/E
(T
)
⊆
Π
E
of
Π
T
via
p
E/E
:
Π
E
Π
E
is
an
E
-tripod
of
Π
n
,
and,
moreover,
for
every
E
E
,
the
image
of
the
E
-tripod
p
Π
E/E
(T
)
via
Π
p
E
/E
:
Π
E
Π
E
is
not
a
tripod
of
Π
n
.
(iv)
Let
h
be
a
positive
integer.
Then
we
shall
say
that
an
E-
tripod
T
of
Π
n
[cf.
(i)]
is
h-descendable
if
there
exists
a
subset
E
⊆
E
such
that
the
image
of
T
via
p
Π
E/E
:
Π
E
Π
E
is
an
E
-
tripod
of
Π
n
,
and,
moreover,
#E
≤
n
−
h.
[Thus,
one
verifies
immediately
that
an
E-tripod
T
⊆
Π
E
of
Π
n
is
1-descendable
if
and
only
if
either
E
=
{1,
·
·
·
,
n}
or
T
fails
to
be
E-strict
—
cf.
(iii).]
Remark
3.3.1.
In
the
notation
of
Definition
3.1,
let
v
∈
Vert(G
i∈E,x
)
be
an
E-tripod
of
X
n
log
[cf.
Definition
3.1,
(v)]
and
T
⊆
Π
E
an
E-tripod
of
Π
n
associated
to
v
[cf.
Definition
3.3,
(i)].
Write
F
v
for
the
irreducible
component
of
the
geometric
fiber
of
p
E/(E\{i})
:
X
E
→
X
E\{i}
at
x
E\{i}
corresponding
to
v;
F
v
log
for
the
log
scheme
obtained
by
equipping
F
v
with
the
log
structure
induced
by
the
log
structure
of
X
E
log
;
n
v
for
the
rank
of
the
group-characteristic
of
F
v
log
[cf.
[MzTa],
Definition
5.1,
(i)]
at
the
generic
point
of
F
v
.
Then
it
is
immediate
that
the
n
v
-interior
U
v
⊆
F
v
of
F
v
log
[cf.
[MzTa],
Definition
5.1,
(i)]
is
a
nonempty
open
subset
of
F
v
which
is
isomorphic
to
P
1
k
\
{0,
1,
∞}
over
k.
Moreover,
one
verifies
easily
that
if
we
write
U
v
log
for
the
log
scheme
obtained
by
equipping
U
v
with
the
log
structure
induced
by
the
log
structure
of
X
E
log
,
then
the
natural
morphism
U
v
log
→
U
v
[obtained
by
forgetting
the
log
∼
structure
of
U
v
log
]
determines
a
natural
outer
isomorphism
T
→
π
1
Σ
(U
v
)
—
where
we
write
“π
1
Σ
(−)”
for
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
of
“(−)”.
In
particular,
we
obtain
a
natural
outer
isomorphism
∼
T
−→
π
1
Σ
(P
1
k
\
{0,
1,
∞})
that
is
well-defined
up
to
composition
with
an
outomorphism
of
π
1
Σ
(P
1
k
\
{0,
1,
∞})
that
arises
from
an
automorphism
of
P
1
k
\
{0,
1,
∞}
over
k.
Definition
3.4.
Let
E
⊆
{1,
·
·
·
,
n}.
(i)
Let
T
⊆
Π
E
be
an
E-tripod
of
Π
n
[cf.
Definition
3.3,
(i)].
Then
T
may
be
regarded
as
the
“Π
1
”
that
occurs
in
the
case
56
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
where
we
take
“X
log
”
to
be
the
smooth
log
curve
associated
to
P
1
k
\
{0,
1,
∞}
[cf.
Remark
3.3.1].
We
shall
write
Out
C
(T
)
⊆
Out(T
)
for
the
[closed]
subgroup
of
Out(T
)
consisting
of
C-admissible
outomorphisms
of
T
[cf.
[CmbCsp],
Definition
1.1,
(ii)];
Out
C
(T
)
cusp
⊆
Out
C
(T
)
for
the
[closed]
subgroup
of
Out(T
)
consisting
of
C-admissible
outomorphisms
of
T
that
induce
the
identity
automorphism
of
the
set
of
T
-conjugacy
classes
of
cuspidal
inertia
subgroups;
Out(T
)
Δ
⊆
Out(T
)
for
the
centralizer
of
the
subgroup
[≃
S
3
,
where
we
write
S
3
for
the
symmetric
group
on
3
letters]
of
Out(T
)
consisting
of
the
outer
modular
symmetries
[cf.
[CmbCsp],
Definition
1.1,
(vi)];
Out(T
)
+
⊆
Out(T
)
for
the
[closed]
subgroup
of
Out(T
)
given
by
the
image
of
the
natural
homomorphism
Out
F
(T
2
)
=
Out
FC
(T
2
)
→
Out(T
)
[cf.
Theorem
2.3,
(ii);
[CmbCsp],
Proposition
1.2,
(iii)]
—
where
we
write
T
2
for
the
“Π
2
”
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
the
smooth
log
curve
associated
to
P
1
k
\
{0,
1,
∞};
def
Out
C
(T
)
Δ
=
Out
C
(T
)
∩
Out(T
)
Δ
;
def
Out
C
(T
)
Δ+
=
Out
C
(T
)
Δ
∩
Out(T
)
+
[cf.
[CmbCsp],
Definition
1.11,
(i)].
(ii)
Let
E
⊆
{1,
·
·
·
,
n};
let
T
⊆
Π
E
,
T
⊆
Π
E
be
E-,
E
-tripods
of
Π
n
[cf.
Definition
3.3,
(i)],
respectively.
Then
we
shall
say
that
∼
an
outer
isomorphism
α
:
T
→
T
is
geometric
if
the
composite
∼
α
∼
∼
π
1
Σ
(P
1
k
\
{0,
1,
∞})
←−
T
−→
T
−→
π
1
Σ
(P
1
k
\
{0,
1,
∞})
—
where
the
first
and
third
arrows
are
natural
outer
isomor-
phisms
of
the
sort
discussed
in
Remark
3.3.1
—
arises
from
an
automorphism
of
P
1
k
\
{0,
1,
∞}
over
k.
Remark
3.4.1.
In
the
notation
of
Definition
3.4,
(ii),
one
verifies
easily
∼
that
every
geometric
outer
isomorphism
α
:
T
→
T
preserves
cuspidal
inertia
subgroups
and
outer
modular
symmetries
[cf.
[CmbCsp],
Defi-
∼
nition
1.1,
(vi)],
and,
moreover,
lifts
to
an
outer
isomorphism
T
2
→
T
2
[i.e.,
of
the
corresponding
“Π
2
’s”]
that
arises
from
an
isomorphism
of
COMBINATORIAL
ANABELIAN
TOPICS
II
57
two-dimensional
configuration
spaces.
In
particular,
the
isomorphism
∼
Out(T
)
→
Out(T
)
induced
by
α
determines
isomorphisms
∼
∼
Out
C
(T
)
−→
Out
C
(T
)
,
Out
C
(T
)
cusp
−→
Out
C
(T
)
cusp
,
∼
∼
Out(T
)
Δ
−→
Out(T
)
Δ
,
Out(T
)
+
−→
Out(T
)
+
[cf.
Definition
3.4,
(i)].
Lemma
3.5
(Triviality
of
the
action
on
the
set
of
cusps).
In
the
notation
of
Definition
3.4,
it
holds
that
Out
C
(T
)
Δ
⊆
Out
C
(T
)
cusp
.
Proof.
This
follows
immediately
from
the
[easily
verified]
fact
that
S
3
is
center-free,
together
with
the
various
definitions
involved.
Lemma
3.6
(Vertices,
cusps,
and
nodes
of
various
fibers).
Let
i,
j
∈
E
be
two
distinct
elements
of
a
subset
E
⊆
{1,
·
·
·
,
n};
x
∈
X
n
(k).
Write
z
i,j,x
∈
VCN(G
j∈E\{i},x
)
for
the
element
of
VCN(G
j∈E\{i},x
)
on
which
x
E\{i}
lies,
that
is
to
say:
If
x
E\{i}
is
a
cusp
or
node
of
the
geo-
log
log
metric
fiber
of
the
projection
p
log
(E\{i})/(E\{i,j})
:
X
E\{i}
→
X
E\{i,j}
over
def
x
log
E\{i,j}
corresponding
to
an
edge
e
∈
Edge(G
j∈E\{i},x
),
then
z
i,j,x
=
e;
if
x
E\{i}
is
neither
a
cusp
nor
a
node
of
the
geometric
fiber
of
the
pro-
log
log
log
jection
p
log
(E\{i})/(E\{i,j})
:
X
E\{i}
→
X
E\{i,j}
over
x
E\{i,j}
,
but
lies
on
the
irreducible
component
of
the
geometric
fiber
corresponding
to
a
vertex
def
v
∈
Vert(G
j∈E\{i},x
),
then
z
i,j,x
=
v.
Then
the
following
hold:
(i)
The
automorphism
of
X
E
log
determined
by
permuting
the
factors
labeled
i,
j
induces
natural
bijections
∼
Vert(G
j∈E\{i},x
)
−→
Vert(G
i∈E\{j},x
)
;
∼
Cusp(G
j∈E\{i},x
)
−→
Cusp(G
i∈E\{j},x
)
;
∼
Node(G
j∈E\{i},x
)
−→
Node(G
i∈E\{j},x
)
.
(ii)
Let
us
write
c
diag
i,j,x
∈
Cusp(G
i∈E,x
)
for
the
diagonal
cusp
of
G
i∈E,x
[cf.
Definition
3.1,
(vi)]
la-
log
beled
j
∈
E\{i}
[cf.
Lemma
3.2,
(ii)].
Then
p
log
E/(E\{j})
:
X
E
→
log
X
E\{j}
induces
a
bijection
∼
Cusp(G
i∈E,x
)
\
{c
diag
i,j,x
}
−→
Cusp(G
i∈E\{j},x
)
.
58
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
log
(iii)
Suppose
that
z
i,j,x
∈
Vert(G
j∈E\{i},x
).
Then
p
log
E/(E\{j})
:
X
E
→
log
X
E\{j}
induces
a
bijection
∼
Vert(G
i∈E,x
)
→
Vert(G
i∈E\{j},x
)
.
(iv)
Suppose
that
z
i,j,x
∈
Edge(G
j∈E\{i},x
).
Then
there
exists
a
unique
vertex
new
v
i,j,x
∈
Vert(G
i∈E,x
)
log
log
such
that
p
log
E/(E\{j})
:
X
E
→
X
E\{j}
induces
a
bijection
∼
new
Vert(G
i∈E,x
)
\
{v
i,j,x
}
→
Vert(G
i∈E\{j},x
)
.
new
new
is
of
type
(0,
3)
[i.e.,
v
i,j,x
is
an
E-tripod
Moreover,
v
i,j,x
diag
log
new
of
X
n
—
cf.
Definition
3.1,
(v)],
and
c
i,j,x
∈
C(v
i,j,x
)
[cf.
new
(ii)].
Finally,
any
verticial
subgroup
of
Π
E
associated
to
v
i,j,x
surjects,
via
p
Π
E/(E\{j})
,
onto
an
edge-like
subgroup
of
Π
E\{j}
associated
to
the
edge
∈
Edge(G
i∈E\{j},x
)
determined
by
z
i,j,x
∈
Edge(G
j∈E\{i},x
)
via
the
bijections
of
(i).
(v)
Suppose
that
#E
=
3.
Write
h
∈
E
\
{i,
j}
for
the
unique
element
of
E
\
{i,
j}.
Suppose,
moreover,
that
z
i,j,x
=
c
diag
j,h,x
∈
Cusp(G
j∈E\{i},x
)
[cf.
(ii)].
Then
the
Π
E
-conjugacy
class
of
new
∈
a
verticial
subgroup
of
Π
E
associated
to
the
vertex
v
i,j,x
Vert(G
i∈E,x
)
[cf.
(iv)]
depends
only
on
i
and
not
on
the
choice
of
the
pair
(j,
x).
Moreover,
these
three
Π
E
-conjugacy
classes
[cf.
the
dependence
on
the
choice
of
i
∈
E]
may
also
be
characterized
uniquely
as
the
Π
E
-conjugacy
classes
of
sub-
groups
of
Π
E
associated
to
some
trigonal
E-tripod
of
Π
n
[cf.
Definition
3.3,
(ii)].
Proof.
First,
we
verify
assertions
(i),
(ii),
(iii),
and
(iv).
To
verify
as-
sertions
(i),
(ii),
(iii),
and
(iv)
—
by
replacing
X
E
log
by
the
base-change
log
log
of
p
log
→
X
E\{i,j}
via
a
suitable
morphism
of
log
schemes
E\{i,j}
:
X
E
log
(Spec
k)
log
→
X
E\{i,j}
whose
image
lies
on
x
E\{i,j}
∈
X
E\{i,j}
(k)
[cf.
Definition
3.1,
(i)]
—
we
may
assume
without
loss
of
generality
that
#E
=
2.
Then
one
verifies
easily
from
the
various
definitions
involved
that
assertions
(i),
(ii),
(iii),
and
(iv)
hold.
This
completes
the
proof
of
assertions
(i),
(ii),
(iii),
and
(iv).
Finally,
we
consider
assertion
(v).
First,
we
observe
the
easily
verified
fact
[cf.
assertions
(iii),
(iv)]
that
the
irreducible
component
corresponding
to
an
E-tripod
of
X
n
log
[cf.
Definition
3.1,
(v)]
that
gives
rise
to
a
trigonal
E-tripod
of
Π
n
neces-
sarily
collapses
to
a
point
upon
projection
to
X
E
for
any
E
⊆
E
of
cardinality
≤
2.
In
light
of
this
observation,
it
follows
immediately
[cf.
assertions
(i),
(ii),
(iii),
(iv)]
that
any
E-tripod
of
X
n
log
that
gives
rise
to
new
”
as
described
in
the
a
trigonal
E-tripod
of
Π
n
arises
as
a
vertex
“v
i,j,x
COMBINATORIAL
ANABELIAN
TOPICS
II
59
statement
of
assertion
(v).
Now
the
remainder
of
assertion
(v)
follows
immediately
from
the
various
definitions
involved
[cf.
also
the
situa-
tion
discussed
in
[CmbCsp],
Definition
1.8,
Proposition
1.9,
Corollary
1.10,
as
well
as
the
discussion,
concerning
specialization
isomorphisms,
preceding
[CmbCsp],
Definition
2.1;
[CbTpI],
Remark
5.6.1].
This
com-
pletes
the
proof
of
Lemma
3.6.
Definition
3.7.
Let
E
⊆
{1,
·
·
·
,
n}.
(i)
Let
v
be
an
E-tripod
of
X
n
log
[cf.
Definition
3.1,
(v)];
thus,
v
belongs
to
Vert(G
i∈E,x
)
for
some
choice
of
i
∈
E
and
x
∈
X
n
(k).
Let
j
∈
E
\
{i}
and
e
∈
Edge(G
j∈E\{i},x
).
Then
we
shall
say
that
v,
or
equivalently,
an
E-tripod
of
Π
n
associated
to
v
[cf.
Definition
3.3,
(i)],
arises
from
e
if
e
=
z
i,j,x
[cf.
the
statement
new
[cf.
Lemma
3.6,
(iv)].
of
Lemma
3.6],
and
v
=
v
i,j,x
(ii)
Let
i
∈
E.
Then
we
shall
say
that
an
E-tripod
of
Π
n
is
i-central
if
#E
=
3,
and,
moreover,
the
tripod
is
a
verticial
subgroup
of
the
sort
discussed
in
Lemma
3.6,
(v),
i.e.,
the
unique,
up
to
Π
E
-conjugacy,
trigonal
E-tripod
of
Π
n
contained
in
Π
E/(E\{i})
[cf.
the
final
portion
of
Lemma
3.6,
(iv)].
We
shall
say
that
an
E-tripod
of
Π
n
is
central
if
it
is
j-central
for
some
j
∈
E.
Remark
3.7.1.
Let
E
⊆
{1,
·
·
·
,
n};
T
⊆
Π
E
an
E-tripod
of
Π
n
[cf.
Definition
3.3,
(i)];
σ
∈
S
n
⊆
Out(Π
n
)
[cf.
the
discussion
at
the
beginning
of
the
present
§3];
σ
∈
Aut(Π
n
)
a
lifting
of
σ
∈
S
n
⊆
Out(Π
n
).
Write
T
σ
⊆
Π
σ(E)
∼
for
the
image
of
T
⊆
Π
E
by
the
isomorphism
Π
E
→
Π
σ(E)
determined
by
σ
∈
Aut(Π
n
).
(i)
One
verifies
easily
that
T
σ
⊆
Π
σ(E)
is
a
σ(E)-tripod
of
Π
n
.
(ii)
If,
moreover,
the
equality
#E
=
3
holds,
and
T
is
i-central
[cf.
Definition
3.7,
(ii)]
for
some
i
∈
E,
then
one
verifies
easily
from
Lemma
3.6,
(v),
that
T
σ
⊆
Π
σ(E)
is
σ(i)-central.
(iii)
In
the
situation
of
(ii),
let
T
⊆
Π
E
be
a
central
E-tripod
of
Π
n
.
Then
it
follows
from
Lemma
3.6,
(v),
that
there
exist
an
element
τ
∈
S
n
⊆
Out(Π
n
)
and
a
lifting
τ
∈
Aut(Π
n
)
of
τ
such
that
τ
preserves
the
subset
E
⊆
{1,
·
·
·
,
n},
and,
moreover,
the
image
of
T
⊆
Π
E
by
the
automorphism
of
Π
E
determined
by
τ
∈
Aut(Π
n
)
coincides
with
T
⊆
Π
E
.
60
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Lemma
3.8
(Strict
tripods).
Let
E
⊆
{1,
·
·
·
,
n}
and
T
⊆
Π
E
an
E-
tripod
of
Π
n
[cf.
Definition
3.3,
(i)]
that
arises
as
a
verticial
subgroup
associated
to
a
vertex
v
∈
Vert(G
i∈E,x
)
for
some
i
∈
{1,
·
·
·
,
n}
[which
thus
implies
that
T
⊆
Π
E/(E\{i})
⊆
Π
E
].
Then
the
following
hold:
(i)
There
exists
a
[not
necessarily
unique!]
subset
E
⊆
E
such
that
T
is
E
-strict
[cf.
Definition
3.3,
(iii)].
In
this
situation,
i
∈
E,
and,
moreover,
p
Π
E/E
:
Π
E
Π
E
induces
an
isomor-
∼
phism
T
→
T
E
onto
an
E
-tripod
T
E
of
Π
n
.
(ii)
T
is
E-strict
if
and
only
if
one
of
the
following
conditions
is
satisfied:
(1)
#E
=
1.
(2
C
)
#E
=
2;
T
⊆
Π
E
is
a
verticial
subgroup
of
Π
E
associated
new
∈
Vert(G
i∈E,x
)
of
Lemma
3.6,
(iv),
for
to
the
vertex
v
i,j,x
some
choice
of
(i,
j,
x)
such
that
z
i,j,x
∈
Cusp(G
j∈E\{i},x
).
[In
particular,
T
arises
from
z
i,j,x
∈
Cusp(G
j∈E\{i},x
)
—
cf.
Definition
3.7,
(i).]
(2
N
)
#E
=
2;
T
⊆
Π
E
is
a
verticial
subgroup
of
Π
E
associated
new
∈
Vert(G
i∈E,x
)
of
Lemma
3.6,
(iv),
for
to
the
vertex
v
i,j,x
some
choice
of
(i,
j,
x)
such
that
z
i,j,x
∈
Node(G
j∈E\{i},x
).
[In
particular,
T
arises
from
z
i,j,x
∈
Node(G
j∈E\{i},x
)
—
cf.
Definition
3.7,
(i).]
(3)
#E
=
3,
and
T
is
central
[cf.
Definition
3.7,
(ii)].
(iii)
Suppose
that
T
is
trigonal
[cf.
Definition
3.3,
(ii)].
Then
there
exists
a
[not
necessarily
unique!]
subset
E
⊆
E
such
that
#E
=
3,
and,
moreover,
the
image
of
T
⊆
Π
E
via
p
Π
E/E
:
Π
E
Π
E
is
a
central
tripod.
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved
by
induction
on
#E
,
together
with
the
well-known
elemen-
tary
fact
that
any
surjective
endomorphism
of
a
topologically
finitely
generated
profinite
group
is
necessarily
bijective.
Next,
we
verify
as-
sertion
(ii).
First,
let
us
observe
that
sufficiency
is
immediate.
Thus,
it
remains
to
verify
necessity.
Suppose
that
T
is
E-strict.
Now
one
verifies
easily
that
if
there
exists
an
element
j
∈
E
\
{i}
such
that
c
diag
i,j,x
∈
C(v)
[cf.
Lemma
3.6,
(ii)],
then
it
follows
immediately
that
the
image
of
T
⊆
Π
E
via
p
Π
E/(E\{j})
:
Π
E
Π
E\{j}
is
an
(E
\
{j})-tripod
[cf.
also
Lemma
3.6,
(iii),
(iv)].
Thus,
since
T
is
E-strict,
we
conclude
that
every
cusp
of
G
i∈E,x
that
is
∈
C(v)
is
non-diagonal.
In
particular,
since
v
is
of
type
(0,
3),
it
follows
immediately
from
Lemma
3.2,
(ii),
that
0
≤
#E
−
1
≤
#C(v)
≤
3.
If
#C(v)
=
0,
then
it
follows
from
the
inequality
#E
−1
≤
#C(v)
that
#E
=
1,
i.e.,
condition
(1)
is
satisfied.
If
#C(v)
=
3,
then
one
verifies
easily
that
#E
=
1,
i.e.,
condition
(1)
COMBINATORIAL
ANABELIAN
TOPICS
II
61
is
satisfied.
Thus,
it
remains
to
verify
assertion
(ii)
in
the
case
where
#C(v)
∈
{1,
2}.
Suppose
that
#C(v)
=
1
and
#E
=
1.
Then
it
follows
immediately
from
the
inequality
#E−1
≤
#C(v)
that
#E
=
2.
Now
let
us
recall
[cf.
Lemma
3.2,
(ii)]
that
the
number
of
diagonal
cusps
of
G
i∈E,x
is
=
#E
−
1
=
1.
Moreover,
the
unique
cusp
on
v
is
the
unique
diagonal
cusp
of
G
i∈E,x
[cf.
the
argument
of
the
preceding
paragraph].
Thus,
one
verifies
easily
that
T
satisfies
condition
(2
N
).
Next,
suppose
that
#C(v)
=
2
and
#E
=
1.
Then
it
follows
immediately
from
the
inequality
#E−1
≤
#C(v)
that
#E
∈
{2,
3}.
Now
let
us
recall
[cf.
Lemma
3.2,
(ii)]
that
if
#E
=
2
(respectively,
#E
=
3),
then
the
number
of
diagonal
cusps
of
G
i∈E,x
is
=
#E−1,
i.e.,
1
(respectively,
2).
Moreover,
the
set
of
diagonal
cusp(s)
of
G
i∈E,x
is
contained
in
(respectively,
is
equal
to)
C(v)
[cf.
the
argument
of
the
preceding
paragraph].
Thus,
one
verifies
easily
that
T
satisfies
condition
(2
C
)
(respectively,
(3)).
This
completes
the
proof
of
assertion
(ii).
Finally,
we
verify
assertion
(iii).
It
follows
from
assertion
(i)
that
there
exists
a
subset
E
⊆
E
such
that
T
is
E
-strict.
Moreover,
it
follows
immediately
from
the
definition
of
a
trigonal
tripod
that
the
E
-tripod
given
by
the
image
p
Π
E/E
(T
)
⊆
Π
E
is
trigonal.
On
the
other
hand,
if
the
E
-tripod
p
Π
E/E
(T
)
satisfies
any
of
conditions
(1),
(2
C
),
(2
N
)
of
assertion
(ii),
then
one
verifies
easily
that
p
Π
E/E
(T
)
is
not
trigonal
[cf.
Π
the
final
portion
of
Lemma
3.6,
(iv)].
Thus,
p
E/E
(T
)
satisfies
condition
(3)
of
assertion
(ii);
in
particular,
p
Π
E/E
(T
)
is
central.
This
completes
the
proof
of
assertion
(iii).
Lemma
3.9
(Generalities
on
normalizers
and
commensura-
tors).
Let
G
be
a
profinite
group,
N
⊆
G
a
normal
closed
subgroup
of
G,
and
H
⊆
G
a
closed
subgroup
of
G.
Then
the
following
hold:
(i)
It
holds
that
C
G
(H)
⊆
C
G
(H
∩
N
).
(ii)
It
holds
that
C
G
(H)
⊆
N
G
(Z
G
loc
(H))
[cf.
the
discussion
entitled
“Topological
groups”
in
“Notations
and
Conventions”].
(iii)
Suppose
that
H
⊆
N
.
Then
it
holds
that
C
G
(H)
⊆
N
G
(C
N
(H)).
In
particular,
if,
moreover,
H
is
commensurably
terminal
in
N
,
then
it
holds
that
C
G
(H)
=
N
G
(H).
def
def
(iv)
Write
H
=
H/(H
∩
N
)
⊆
G
=
G/N
.
If
H
∩
N
is
commen-
surably
terminal
in
N
,
and
the
image
of
C
G
(H)
⊆
G
in
G
is
contained
in
N
G
(H),
then
C
G
(H)
=
N
G
(H).
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved.
Next,
we
verify
assertion
(ii).
Let
g
∈
C
G
(H)
and
a
∈
Z
G
loc
(H).
Since
Z
G
loc
(H)
=
Z
G
loc
(H
∩
(g
−1
·
H
·
g))
=
Z
G
loc
(g
−1
·
H
·
g),
62
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
there
exists
an
open
subgroup
U
⊆
H
of
H
such
that
a
∈
Z
G
(g
−1
·U
·g).
But
this
implies
that
gag
−1
∈
Z
G
(U
)
⊆
Z
G
loc
(H).
This
completes
the
proof
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
Let
g
∈
C
G
(H)
and
a
∈
C
N
(H).
Since
C
N
(H)
⊆
C
G
(H)
=
C
G
(H
∩
(g
−1
·
H
·
g))
=
C
G
(g
−1
·
H
·
g),
we
conclude
that
ag
−1
·
H
·
ga
−1
is
commensurate
with
g
−1
·
H
·
g.
In
particular,
gag
−1
·
H
·
ga
−1
g
−1
is
commensurate
with
H,
i.e.,
gag
−1
∈
C
G
(H)
∩
N
=
C
N
(H).
This
completes
the
proof
of
assertion
(iii).
Finally,
we
verify
assertion
(iv).
First,
we
observe
that
since
H
∩
N
is
commensurably
terminal
in
N
,
one
verifies
easily
that
H
=
N
H·N
(H
∩
N
).
Let
g
∈
C
G
(H).
Then
since
the
image
of
C
G
(H)
⊆
G
in
G
is
contained
in
N
G
(H),
it
is
immediate
that
g
·
H
·
g
−1
⊆
H
·
N
.
On
the
other
hand,
again
by
applying
the
fact
that
H
∩
N
is
commensurably
terminal
in
N
,
we
conclude
immediately
from
assertions
(i),
(iii),
that
C
G
(H)
⊆
C
G
(H
∩
N
)
=
N
G
(H
∩
N
).
Thus,
we
obtain
that
(g
·
H
·
g
−1
)
∩
N
=
H
∩
N
;
in
particular,
g
·
H
·
g
−1
⊆
N
H·N
((g
·
H
·
g
−1
)
∩
N
)
=
N
H·N
(H
∩
N
)
=
H,
i.e.,
g
∈
N
G
(H).
This
completes
the
proof
of
assertion
(iv).
Lemma
3.10
(Restrictions
of
outomorphisms).
Let
G
be
a
profi-
nite
group
and
H
⊆
G
a
closed
subgroup
of
G.
Write
Out
H
(G)
⊆
Out(G)
for
the
group
of
outomorphisms
of
G
that
preserve
the
G-
conjugacy
class
of
H.
Suppose
that
the
homomorphism
N
G
(H)
→
Aut(H)
determined
by
conjugation
factors
through
Inn(H)
⊆
Aut(H).
Then
the
following
hold:
(i)
For
α
∈
Out
H
(G),
let
us
write
α|
H
for
the
outomorphism
of
H
determined
by
the
restriction
to
H
⊆
G
of
a
lifting
α
∈
Aut(G)
of
α
such
that
α
(H)
=
H.
Then
α|
H
does
not
depend
on
the
choice
of
the
lifting
“
α
”,
and
the
map
Out
H
(G)
−→
Out(H)
given
by
assigning
α
→
α|
H
is
a
group
homomorphism.
(ii)
The
homomorphism
Out
H
(G)
−→
Out(H)
of
(i)
depends
only
on
the
G-conjugacy
class
of
the
closed
def
subgroup
H
⊆
G,
i.e.,
if
we
write
H
γ
=
γ
·
H
·
γ
−1
for
γ
∈
G,
then
the
diagram
Out
H
(G)
−−−→
Out(H)
⏐
⏐
γ
Out
H
(G)
−−−→
Out(H
γ
)
—
where
the
upper
(respectively,
lower)
horizontal
arrow
is
the
homomorphism
given
by
mapping
α
→
α|
H
(respectively,
COMBINATORIAL
ANABELIAN
TOPICS
II
63
α
→
α|
H
γ
),
and
the
right-hand
vertical
arrow
is
the
isomor-
∼
phism
obtained
by
conjugation
via
the
isomorphism
H
→
H
γ
determined
by
conjugation
by
γ
∈
G
—
commutes.
Proof.
Assertion
(i)
follows
immediately
from
our
assumption
that
the
homomorphism
N
G
(H)
→
Aut(H)
determined
by
conjugation
factors
through
Inn(H)
⊆
Aut(H),
together
with
the
various
definitions
in-
volved.
Assertion
(ii)
follows
immediately
from
the
various
definitions
involved.
This
completes
the
proof
of
Lemma
3.10.
Lemma
3.11
(Commensurator
of
a
tripod
arising
from
an
edge).
In
the
notation
of
Lemma
3.6,
suppose
that
(j,
i)
=
(1,
2);
E
=
{i,
j};
z
i,j,x
∈
Edge(G
j∈E\{i},x
).
[Thus,
G
j∈E\{i},x
=
G
i∈E\{j},x
=
G;
∼
∼
Π
2
=
Π
E
;
Π
1
=
Π
{j}
→
Π
G
j∈E\{i},x
=
Π
G
;
Π
2/1
=
Π
E/(E\{i})
→
Π
G
i∈E,x
.]
def
def
def
def
Π
Write
G
2/1
=
G
i∈E,x
;
G
1\2
=
G
j∈E,x
;
p
Π
1\2
=
p
E/{2}
:
Π
2
Π
{2}
;
Π
1\2
=
∼
def
def
diag
=
c
diag
Ker(p
Π
i,j,x
∈
1\2
)
=
Π
E/{2}
→
Π
G
1\2
;
z
x
=
z
i,j,x
∈
Edge(G);
c
def
def
new
new
=
v
i,j,x
∈
Vert(G
2/1
)
Cusp(G
2/1
)
[cf.
Lemma
3.6,
(ii)];
v
new
=
v
2/1
new
[cf.
Lemma
3.6,
(iv)];
v
1\2
∈
Vert(G
1\2
)
for
the
vertex
that
corresponds
∼
to
v
new
∈
Vert(G
2/1
)
via
the
natural
bijection
Vert(G
2/1
)
→
Vert(G
1\2
)
induced
by
the
automorphism
of
X
E
log
determined
by
permuting
the
fac-
tors
labeled
i,
j;
Y
→
X
E
for
the
base-change
—
by
the
morphism
log
X
E
→
X
{1}
×
k
X
{2}
=
X
×
k
X
determined
by
p
log
E/{1}
and
p
E/{2}
—
of
the
geometric
point
of
X
{1}
×
k
X
{2}
=
X
×
k
X
determined
by
the
geomet-
ric
points
x
{1}
of
X
{1}
=
X
and
x
{1}
of
X
{2}
=
X
of
Definition
3.1,
(i)
[i.e.,
as
opposed
to
the
geometric
point
of
X
{1}
×
k
X
{2}
=
X
×
k
X
deter-
mined
by
the
geometric
points
x
{1}
of
X
{1}
=
X
and
x
{2}
of
X
{2}
=
X];
Y
log
for
the
log
scheme
obtained
by
equipping
Y
with
the
log
structure
induced
by
the
log
structure
of
X
E
log
;
U
⊆
Y
for
the
2-interior
of
Y
log
[cf.
[MzTa],
Definition
5.1,
(i)];
U
log
for
the
log
scheme
obtained
by
equipping
U
with
the
log
structure
induced
by
the
log
structure
of
X
E
log
;
Π
U
for
the
maximal
pro-Σ
quotient
of
the
kernel
of
the
natural
sur-
jection
π
1
(U
log
)
π
1
((Spec
k)
log
).
[Thus,
one
verifies
easily
that
Y
is
isomorphic
to
P
1
k
;
that
the
complement
Y
\U
consists
of
three
closed
new
new
and
v
1\2
correspond
to
the
closed
points
of
Y
;
that
the
vertices
v
2/1
irreducible
subscheme
Y
⊆
X
E
;
and
that
the
point
corresponding
to
the
cusp
c
diag
is
contained
in
Y
—
cf.
Lemma
3.6,
(iv).]
Let
Π
z
x
⊆
Π
1
be
an
edge-like
subgroup
associated
to
z
x
∈
Edge(G);
Π
c
diag
⊆
Π
2/1
∩
Π
1\2
a
cuspidal
subgroup
associated
to
c
diag
;
Π
v
new
⊆
Π
2/1
a
verticial
sub-
def
new
=
Π
v
new
;
group
associated
to
v
new
that
contains
Π
c
diag
⊆
Π
2
;
Π
v
2/1
new
new
⊆
Π
1\2
a
verticial
subgroup
associated
to
v
Π
v
1\2
1\2
that
contains
def
def
Π
c
diag
⊆
Π
2
.
Write
Π
2
|
z
x
=
Π
2
×
Π
1
Π
z
x
⊆
Π
2
;
D
c
diag
=
N
Π
2
(Π
c
diag
);
64
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
def
def
I
v
new
|
z
x
=
Z
Π
2
|
zx
(Π
v
new
)
⊆
D
v
new
|
z
x
=
N
Π
2
|
zx
(Π
v
new
).
Then
the
follow-
ing
hold:
(i)
It
holds
that
D
c
diag
∩
Π
2/1
=
D
c
diag
∩
Π
1\2
=
C
Π
2
(Π
c
diag
)
∩
Π
2/1
=
C
Π
2
(Π
c
diag
)
∩
Π
1\2
=
Π
c
diag
.
(ii)
It
holds
that
C
Π
2
(Π
c
diag
)
=
D
c
diag
.
Π
(iii)
The
surjections
p
Π
2/1
:
Π
2
Π
1
,
p
1\2
:
Π
2
Π
{2}
determine
∼
∼
isomorphisms
D
c
diag
/Π
c
diag
→
Π
1
,
D
c
diag
/Π
c
diag
→
Π
{2}
,
re-
spectively,
such
that
the
resulting
composite
outer
isomorphism
∼
Π
1
→
Π
{2}
is
the
identity
outer
isomorphism.
(iv)
The
natural
inclusions
Π
v
new
,
I
v
new
|
z
x
→
D
v
new
|
z
x
determine
∼
an
isomorphism
Π
v
new
×
I
v
new
|
z
x
→
D
v
new
|
z
x
=
C
Π
2
|
zx
(Π
v
new
).
Moreover,
the
composite
I
v
new
|
z
x
→
D
v
new
|
z
x
→
Π
z
x
is
an
iso-
morphism.
(v)
It
holds
that
C
Π
2
(D
v
new
|
z
x
)
⊆
C
Π
2
(Π
v
new
).
(vi)
D
v
new
|
z
x
is
commensurably
terminal
in
Π
2
,
i.e.,
it
holds
that
D
v
new
|
z
x
=
C
Π
2
(D
v
new
|
z
x
).
(vii)
It
holds
that
Z
Π
2
(Π
v
new
)
=
Z
Π
loc
2
(Π
v
new
)
=
I
v
new
|
z
x
.
Moreover,
these
profinite
groups
are
isomorphic
to
Z
Σ
[cf.
the
discussion
entitled
“Numbers”
in
[CbTpI],
§0].
(viii)
It
holds
that
C
Π
2
(Π
v
new
)
=
D
v
new
|
z
x
=
Π
v
new
×
Z
Π
2
(Π
v
new
).
In
particular,
the
equality
C
Π
2
(Π
v
new
)
=
N
Π
2
(Π
v
new
)
holds.
(ix)
It
holds
that
Z(C
Π
2
(Π
v
new
))
=
Z
Π
2
(Π
v
new
).
(x)
It
holds
that
new
)
∩
Π
2/1
=
Π
v
new
,
C
Π
2
(Π
v
2/1
2/1
new
)
∩
Π
1\2
=
Π
v
new
,
C
Π
2
(Π
v
1\2
1\2
new
)
=
C
Π
(Π
v
new
)
.
C
Π
2
(Π
v
2/1
2
1\2
Moreover,
for
suitable
choices
of
basepoints
of
the
log
schemes
U
log
and
X
E
log
,
the
natural
morphism
U
log
→
X
E
log
induces
an
∼
new
)
=
C
Π
(Π
v
new
).
isomorphism
Π
U
→
C
Π
2
(Π
v
2/1
2
1\2
Proof.
First,
we
verify
assertion
(i).
Now
it
is
immediate
that
we
have
inclusions
Π
c
diag
⊆
D
c
diag
⊆
C
Π
2
(Π
c
diag
).
In
particular,
since
Π
c
diag
is
commensurably
terminal
in
Π
2/1
and
Π
1\2
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
we
obtain
that
Π
c
diag
⊆
D
c
diag
∩
Π
2/1
⊆
C
Π
2
(Π
c
diag
)
∩
Π
2/1
=
C
Π
2/1
(Π
c
diag
)
=
Π
c
diag
;
Π
c
diag
⊆
D
c
diag
∩
Π
1\2
⊆
C
Π
2
(Π
c
diag
)
∩
Π
1\2
=
C
Π
1\2
(Π
c
diag
)
=
Π
c
diag
.
This
completes
the
proof
of
assertion
(i).
As-
sertions
(ii),
(iii)
follow
immediately
from
assertion
(i),
together
with
COMBINATORIAL
ANABELIAN
TOPICS
II
65
p
Π
2/1
the
[easily
verified]
fact
that
the
composites
D
c
diag
→
Π
2
Π
1
and
p
Π
1\2
D
c
diag
→
Π
2
Π
{2}
are
surjective.
Next,
we
verify
assertion
(iv).
It
follows
immediately
from
the
vari-
ous
definitions
involved
—
by
considering
a
suitable
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
and
applying
a
suitable
specialization
iso-
morphism
[cf.
the
discussion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1]
—
that,
to
verify
assertion
(iv),
we
may
assume
without
loss
of
generality
that
Cusp(G)
∪
{z
x
}
=
Edge(G).
Then,
in
light
of
the
well-known
local
structure
of
X
log
in
a
neigh-
borhood
of
the
node
or
cusp
corresponding
to
z
x
,
one
verifies
easily
∼
that
the
outer
action
Π
z
x
→
Out(Π
2/1
)
→
Out(Π
G
2/1
)
arising
from
the
natural
exact
sequence
1
−→
Π
2/1
−→
Π
2
|
z
x
−→
Π
z
x
−→
1
is
of
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)],
hence,
in
partic-
ular,
that
the
composite
I
v
new
|
z
x
→
D
v
new
|
z
x
→
Π
z
x
is
an
isomor-
phism.
Thus,
assertion
(iv)
follows
immediately
from
[NodNon],
Re-
mark
2.7.1,
together
with
the
commensurable
terminality
of
Π
v
new
in
Π
2/1
[cf.
[CmbGC],
Proposition
1.2,
(ii)]
and
the
fact
that
the
compos-
ite
D
v
new
|
z
x
→
Π
2
|
z
x
Π
z
x
is
surjective.
This
completes
the
proof
of
assertion
(iv).
Next,
we
verify
assertion
(v).
It
follows
immediately
from
asser-
tion
(iv),
together
with
the
commensurable
terminality
of
Π
v
new
in
Π
2/1
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
that
D
v
new
|
z
x
∩
Π
2/1
=
Π
v
new
.
Thus,
since
Π
2/1
is
normal
in
Π
2
,
assertion
(v)
follows
immediately
from
Lemma
3.9,
(i).
This
completes
the
proof
of
assertion
(v).
Next,
we
verify
assertion
(vi).
Since
the
image
of
the
composite
p
Π
2/1
D
v
new
|
z
x
→
Π
2
Π
1
coincides
with
Π
z
x
⊆
Π
1
[cf.
assertion
(iv)],
and
Π
z
x
⊆
Π
1
is
commensurably
terminal
in
Π
1
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
it
follows
immediately
that
C
Π
2
(D
v
new
|
z
x
)
⊆
Π
2
|
z
x
.
In
partic-
ular,
it
follows
immediately
from
assertions
(iv),
(v)
that
D
v
new
|
z
x
⊆
C
Π
2
(D
v
new
|
z
x
)
⊆
C
Π
2
(Π
v
new
)
∩
Π
2
|
z
x
=
C
Π
2
|
zx
(Π
v
new
)
=
D
v
new
|
z
x
.
This
completes
the
proof
of
assertion
(vi).
Next,
we
verify
assertion
(vii).
It
follows
from
assertion
(iv)
and
[CmbGC],
Remark
1.1.3,
that
I
v
new
|
z
x
is
isomorphic
to
Z
Σ
.
Moreover,
it
follows
from
the
various
definitions
involved
that
we
have
inclusions
I
v
new
|
z
x
⊆
Z
Π
2
(Π
v
new
)
⊆
Z
Π
loc
2
(Π
v
new
).
Thus,
to
verify
assertion
(vii),
it
suffices
to
verify
that
Z
Π
loc
2
(Π
v
new
)
⊆
I
v
new
|
z
x
.
To
this
end,
let
us
ob-
serve
that
it
follows
immediately
from
the
final
portion
of
Lemma
3.6,
∼
new
)
⊆
Π
{2}
→
Π
G
is
an
edge-like
subgroup
(iv),
that
the
image
p
Π
1\2
(Π
v
∼
of
Π
{2}
→
Π
G
associated
to
z
x
∈
Edge(G).
Thus,
since
every
edge-
like
subgroup
is
commensurably
terminal
[cf.
[CmbGC],
Proposition
66
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
loc
new
))
⊆
Π
{2}
→
Π
G
1.2,
(ii)],
it
follows
that
the
image
p
Π
1\2
(Z
Π
2
(Π
v
∼
is
contained
in
an
edge-like
subgroup
of
Π
{2}
→
Π
G
associated
to
z
x
∈
Edge(G).
On
the
other
hand,
since
Π
c
diag
⊆
Π
v
new
,
we
have
Z
Π
loc
2
(Π
v
new
)
⊆
Z
Π
loc
2
(Π
c
diag
)
⊆
C
Π
2
(Π
c
diag
)
=
D
c
diag
[cf.
assertion
(ii)].
In
particular,
it
follows
immediately
from
assertion
(iii),
together
with
the
fact
[cf.
assertion
(iv)]
that
I
v
new
|
z
x
⊆
Z
Π
loc
2
(Π
v
new
)
surjects
onto
loc
new
))
⊆
Π
1
is
Π
z
x
[cf.
also
[NodNon],
Lemma
1.5],
that
p
Π
2/1
(Z
Π
2
(Π
v
loc
contained
in
Π
z
x
⊆
Π
1
,
i.e.,
Z
Π
2
(Π
v
new
)
⊆
Π
2
|
z
x
.
Thus,
it
follows
im-
mediately
from
assertion
(iv),
together
with
the
slimness
of
Π
v
new
[cf.
[CmbGC],
Remark
1.1.3],
that
Z
Π
loc
2
(Π
v
new
)
⊆
I
v
new
|
z
x
.
This
completes
the
proof
of
assertion
(vii).
Next,
we
verify
assertion
(viii).
It
follows
from
assertion
(vii),
to-
gether
with
Lemma
3.9,
(ii),
that
C
Π
2
(Π
v
new
)
⊆
N
Π
2
(I
v
new
|
z
x
).
In
par-
ticular,
since
D
v
new
|
z
x
is
generated
by
Π
v
new
,
I
v
new
|
z
x
[cf.
assertion
(iv)],
it
follows
immediately
that
(D
v
new
|
z
x
⊆)
C
Π
2
(Π
v
new
)
⊆
C
Π
2
(D
v
new
|
z
x
).
Thus,
the
first
equality
of
assertion
(viii)
follows
from
assertion
(vi);
the
second
equality
of
assertion
(viii)
follows
immediately
from
assertions
(iv),
(vii).
This
completes
the
proof
of
assertion
(viii).
Next,
we
verify
assertion
(ix).
Let
us
recall
from
[CmbGC],
Remark
1.1.3,
that
Π
v
new
is
slim.
Thus,
assertion
(ix)
follows
from
assertion
(viii),
together
with
the
final
portion
of
assertion
(vii).
This
completes
the
proof
of
assertion
(ix).
Finally,
we
verify
assertion
(x).
The
first
two
equalities
follow
from
[CmbGC],
Proposition
1.2,
(ii).
Next,
let
us
observe
that
since
[it
is
im-
mediate
that]
the
automorphism
of
X
E
log
determined
by
permuting
the
new
new
factors
labeled
i,
j
stabilizes
U
,
but
permutes
v
2/1
and
v
1\2
,
one
verifies
immediately
that,
to
verify
assertion
(x),
it
suffices
to
verify
that,
for
suitable
choices
of
basepoints
of
the
log
schemes
U
log
and
X
E
log
,
the
nat-
∼
ural
morphism
U
log
→
X
E
log
induces
an
isomorphism
Π
U
→
C
Π
2
(Π
v
new
)
new
new
)).
To
this
end,
let
us
observe
that
since
the
vertex
v
(=
C
Π
2
(Π
v
2/1
corresponds
to
the
closed
irreducible
subscheme
Y
⊆
X
E
[cf.
the
dis-
cussion
following
the
definition
of
Π
U
in
the
statement
of
Lemma
3.11],
it
follows
immediately
from
the
various
definitions
involved
that,
for
suitable
choices
of
basepoints
of
the
log
schemes
U
log
and
X
E
log
,
the
natural
morphism
U
log
→
X
E
log
gives
rise
to
a
commutative
diagram
1
−−−→
Π
U/z
x
−−−→
⏐
⏐
Π
U
⏐
⏐
−−−→
Π
z
x
−−−→
1
1
−−−→
Π
v
new
−−−→
D
v
new
|
z
x
−−−→
Π
z
x
−−−→
1
—
where
we
write
Π
U/z
x
for
the
kernel
of
the
natural
surjection
Π
U
Π
z
x
;
the
horizontal
sequences
are
exact;
the
exactness
of
the
lower
horizontal
sequence
follows
from
assertion
(iv);
the
left-hand
vertical
arrow
is
an
isomorphism.
Thus,
it
follows
from
assertion
(viii)
that,
COMBINATORIAL
ANABELIAN
TOPICS
II
67
for
suitable
choices
of
basepoints
of
the
log
schemes
U
log
and
X
E
log
,
∼
the
natural
morphism
U
log
→
X
E
log
induces
an
isomorphism
Π
U
→
D
v
new
|
z
x
=
C
Π
2
(Π
v
new
),
as
desired.
This
completes
the
proof
of
assertion
(x),
hence
also
of
Lemma
3.11.
The
first
item
of
the
following
result
[i.e.,
Lemma
3.12,
(i)]
is,
along
with
its
proof,
a
routine
generalization
of
[CmbCsp],
Corollary
1.10,
(ii).
Lemma
3.12
(Commensurator
of
a
tripod).
Let
E
⊆
{1,
·
·
·
,
n}
and
T
⊆
Π
E
an
E-tripod
of
Π
n
[cf.
Definition
3.3,
(i)].
Then
the
following
hold:
(i)
It
holds
that
C
Π
E
(T
)
=
T
×Z
Π
E
(T
).
Thus,
if
an
outomorphism
α
of
Π
E
preserves
the
Π
E
-conjugacy
class
of
T
,
then
one
may
define
α|
T
∈
Out(T
)
[cf.
Lemma
3.10,
(i)].
(ii)
Suppose
that
n
=
#E
=
3,
and
that
T
is
central
[cf.
Defi-
nition
3.7,
(ii)].
Let
T
⊆
Π
E
=
Π
n
be
a
central
E-tripod
of
Π
n
.
Then
C
Π
n
(T
)
(respectively,
N
Π
n
(T
);
Z
Π
n
(T
))
is
a
Π
n
-
conjugate
of
C
Π
n
(T
)
(respectively,
N
Π
n
(T
);
Z
Π
n
(T
)).
Proof.
Let
i
∈
E;
x
∈
X
n
(k);
v
∈
Vert(G
i∈E,x
)
be
such
that
v
is
of
type
(0,
3),
and,
moreover,
T
is
a
verticial
subgroup
of
Π
E
associated
to
v
∈
Vert(G
i∈E,x
).
[Thus,
we
have
an
inclusion
T
⊆
Π
E/(E\{i})
⊆
Π
E
—
cf.
Definition
3.1,
(iv).]
First,
we
verify
assertion
(i).
Since
T
⊆
Π
E/(E\{i})
⊆
Π
E
,
and
T
is
commensurably
terminal
in
Π
E/(E\{i})
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
it
follows
from
Lemma
3.9,
(iii),
that
C
Π
E
(T
)
=
N
Π
E
(T
).
Thus,
in
light
of
the
slimness
of
T
[cf.
[CmbGC],
Remark
1.1.3],
to
ver-
ify
assertion
(i),
it
suffices
to
verify
that
the
natural
outer
action
of
N
Π
E
(T
)
on
T
is
trivial.
To
this
end,
let
E
⊆
E
be
such
that
T
is
E
-strict
[cf.
Lemma
3.8,
(i)];
write
T
E
⊆
Π
E
for
the
image
of
T
via
p
Π
E/E
:
Π
E
Π
E
.
Then
it
is
immediate
that
the
image
of
N
Π
E
(T
)
Π
via
p
E/E
:
Π
E
Π
E
is
contained
in
N
Π
E
(T
E
),
and
that
the
natural
surjection
T
T
E
is
an
isomorphism
[cf.
Lemma
3.8,
(i)].
Thus,
one
verifies
easily
—
by
replacing
E,
T
by
E
,
T
E
,
respectively
—
that,
to
verify
that
the
natural
outer
action
of
N
Π
E
(T
)
on
T
is
trivial,
we
may
assume
without
loss
of
generality
that
T
is
E-strict.
If
T
satisfies
condition
(1)
of
Lemma
3.8,
(ii),
then
assertion
(i)
follows
from
the
commensurable
terminality
of
T
in
Π
E
[cf.
[CmbGC],
Proposition
1.2,
(ii)].
If
T
satisfies
either
condition
(2
C
)
or
condition
(2
N
)
of
Lemma
3.8,
(ii),
then
assertion
(i)
follows
immediately
from
Lemma
3.11,
(viii).
If
T
satisfies
condition
(3)
of
Lemma
3.8,
(ii),
then
one
verifies
easily
from
the
various
definitions
involved
—
by
considering
a
suitable
stable
log
68
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
curve
of
type
(g,
r)
over
(Spec
k)
log
and
applying
a
suitable
specializa-
tion
isomorphism
[cf.
the
discussion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1]
—
that,
to
verify
assertion
(i),
we
may
assume
without
loss
of
generality
that
Node(G)
=
∅.
Thus,
assertion
(i)
follows
immediately
from
[CmbCsp],
Corollary
1.10,
(ii).
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Let
us
recall
from
Remark
3.7.1,
(iii),
that
there
exist
an
element
τ
∈
S
3
⊆
Out(Π
3
)
[cf.
the
discussion
at
the
beginning
of
the
present
§3]
and
a
lifting
τ
∈
Aut(Π
3
)
of
τ
such
that
the
image
of
T
⊆
Π
3
by
the
automorphism
τ
∈
Aut(Π
3
)
coincides
with
T
⊆
Π
3
.
Next,
let
us
observe
that
one
verifies
easily
that
τ
∈
S
3
may
be
written
as
a
product
of
transpositions
in
S
3
.
Thus,
in
the
remainder
of
the
proof
of
assertion
(ii),
we
may
assume
without
loss
of
generality
that
τ
is
a
transposition
in
S
3
.
Moreover,
in
the
remainder
of
the
proof
of
assertion
(ii),
we
may
assume
without
loss
of
generality,
by
conjugating
by
a
suitable
element
of
S
3
,
that
τ
is
the
transposition
“(1,
2)”
in
S
3
.
Thus,
if,
moreover,
i
=
3
[i.e.,
the
E-tripod
T
is
3-central],
then
it
follows
from
Lemma
3.6,
(v),
that
T
is
a
Π
3
-
conjugate
of
T
,
hence
that
C
Π
3
(T
)
(respectively,
N
Π
3
(T
);
Z
Π
3
(T
))
is
a
Π
3
-conjugate
of
C
Π
3
(T
)
(respectively,
N
Π
3
(T
);
Z
Π
3
(T
)).
In
particular,
in
the
remainder
of
the
proof
of
assertion
(ii),
we
may
assume
without
loss
of
generality,
by
conjugating
by
τ
∈
S
3
if
necessary,
that
i
=
2,
i.e.,
that
the
E-tripods
T
,
T
are
2-central,
1-central,
respectively.
Next,
let
us
observe
that,
in
this
situation,
one
verifies
immedi-
ately
from
the
various
definitions
involved
that
there
exists
a
natural
identification
between
Π
{1,2,3}/{3}
and
the
“Π
2
”
that
arises
in
the
case
log
log
where
we
take
“X
log
”
to
be
the
base-change
of
p
log
{3}
:
X
{2,3}
→
X
{3}
log
via
a
suitable
morphism
of
log
schemes
(Spec
k)
log
→
X
{3}
.
More-
over,
one
also
verifies
immediately
from
the
various
definitions
involved
[cf.
also
Lemma
3.6,
(v)]
that
this
natural
identification
maps
suitable
Π
3
-conjugates
of
T
,
T
,
respectively,
bijectively
onto
the
closed
sub-
new
”,
“Π
v
new
”
of
the
“Π
2
”
that
appears
in
the
statement
of
groups
“Π
v
2/1
1\2
Lemma
3.11.
In
particular,
it
follows
from
Lemma
3.11,
(viii),
(ix),
(x),
that
the
following
assertions
hold:
(a)
The
following
equalities
hold:
C
Π
{1,2,3}/{3}
(T
)
=
T
×
Z
Π
{1,2,3}/{3}
(T
),
C
Π
{1,2,3}/{3}
(T
)
=
T
×
Z
Π
{1,2,3}/{3}
(T
).
(b)
The
following
equalities
hold:
C
Π
{1,2,3}/{3}
(T
)
∩
Π
{1,2,3}/{1,3}
=
T,
C
Π
{1,2,3}/{3}
(T
)
∩
Π
{1,2,3}/{2,3}
=
T
.
COMBINATORIAL
ANABELIAN
TOPICS
II
69
(c)
The
subgroup
C
Π
{1,2,3}/{3}
(T
)
(respectively,
Z
Π
{1,2,3}/{3}
(T
))
is
a
Π
{1,2,3}/{3}
-conjugate
of
the
subgroup
C
Π
{1,2,3}/{3}
(T
)
(respec-
tively,
Z
Π
{1,2,3}/{3}
(T
)).
In
particular,
it
follows
from
(c)
that,
to
verify
assertion
(ii),
it
suffices
to
verify
the
following
assertion:
Claim
3.12.A:
The
following
equalities
hold:
C
Π
3
(T
)
=
C
Π
3
(C
Π
{1,2,3}/{3}
(T
)),
C
Π
3
(T
)
=
C
Π
3
(C
Π
{1,2,3}/{3}
(T
)),
N
Π
3
(T
)
=
N
Π
3
(C
Π
{1,2,3}/{3}
(T
)),
N
Π
3
(T
)
=
N
Π
3
(C
Π
{1,2,3}/{3}
(T
)),
Z
Π
3
(T
)
=
Z
Π
3
(C
Π
{1,2,3}/{3}
(T
)),
Z
Π
3
(T
)
=
Z
Π
3
(C
Π
{1,2,3}/{3}
(T
)).
First,
we
verify
the
first
four
equalities
of
Claim
3.12.A.
Observe
that
since
Π
{1,2,3}/{3}
is
a
normal
closed
subgroup
of
Π
3
and
contains
both
T
and
T
,
it
follows
from
Lemma
3.9,
(iii),
that
the
inclusions
N
Π
3
(T
)
⊆
C
Π
3
(T
)
⊆
N
Π
3
(C
Π
{1,2,3}/{3}
(T
))
⊆
C
Π
3
(C
Π
{1,2,3}/{3}
(T
)),
N
Π
3
(T
)
⊆
C
Π
3
(T
)
⊆
N
Π
3
(C
Π
{1,2,3}/{3}
(T
))
⊆
C
Π
3
(C
Π
{1,2,3}/{3}
(T
))
hold.
Moreover,
by
the
normality
of
Π
{1,2,3}/{1,3}
and
Π
{1,2,3}/{2,3}
in
Π
3
,
one
verifies
easily,
by
applying
(b),
that
the
inclusions
N
Π
3
(C
Π
{1,2,3}/{3}
(T
))
⊆
N
Π
3
(T
),
C
Π
3
(C
Π
{1,2,3}/{3}
(T
))
⊆
C
Π
3
(T
),
N
Π
3
(C
Π
{1,2,3}/{3}
(T
))
⊆
N
Π
3
(T
),
C
Π
3
(C
Π
{1,2,3}/{3}
(T
))
⊆
C
Π
3
(T
)
hold.
This
completes
the
proof
of
the
first
four
equalities
of
Claim
3.12.A.
Finally,
we
verify
the
final
two
equalities
of
Claim
3.12.A.
Let
us
first
observe
that
the
inclusions
T
⊆
C
Π
{1,2,3}/{3}
(T
),
T
⊆
C
Π
{1,2,3}/{3}
(T
)
imply
that
Z
Π
3
(C
Π
{1,2,3}/{3}
(T
))
⊆
Z
Π
3
(T
),
Z
Π
3
(C
Π
{1,2,3}/{3}
(T
))
⊆
Z
Π
3
(T
).
Thus,
it
follows
immediately
from
(a)
that,
to
verify
the
final
two
equal-
ities
of
Claim
3.12.A,
it
suffices
to
verify
the
following
assertion:
Claim
3.12.B:
The
following
inclusions
hold:
Z
Π
3
(T
)
⊆
Z
Π
3
(Z
Π
{1,2,3}/{3}
(T
)),
Z
Π
3
(T
)
⊆
Z
Π
3
(Z
Π
{1,2,3}/{3}
(T
)).
First,
let
us
observe
that
one
verifies
immediately
from
the
various
def-
initions
involved
that
the
natural
identification
that
appears
in
the
dis-
cussion
preceding
assertion
(a)
in
the
present
proof
of
Lemma
3.12,
(ii),
determines
a
natural
identification
between
Π
{2,3}/{3}
and
the
“Π
1
=
Π
{2}
”
that
arises
in
the
case
where
we
take
“X
log
”
to
be
as
in
the
discussion
preceding
assertion
(a)
in
the
present
proof
of
Lemma
3.12,
(ii).
Thus,
it
follows
immediately
from
the
final
portion
of
Lemma
3.6,
(iv),
that
the
image
J
T
⊆
Π
{2,3}/{3}
of
T
⊆
Π
{1,2,3}/{3}
in
Π
{2,3}/{3}
corresponds,
via
the
natural
identification
just
discussed,
to
an
edge-
like
subgroup
of
“Π
1
=
Π
{2}
”
associated
to
the
edge
z
x
∈
Edge(G)
that
appears
in
the
statement
of
Lemma
3.11.
Moreover,
it
follows
70
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
immediately
from
(c)
and
Lemma
3.11,
(iv),
(vii),
that
the
surjection
Π
{1,2,3}/{3}
Π
{2,3}/{3}
induces
an
isomorphism
∼
Π
{1,2,3}/{3}
⊇
Z
Π
{1,2,3}/{3}
(T
)
→
J
Z
⊆
Π
{2,3}/{3}
—
where
the
closed
subgroup
J
Z
⊆
Π
{2,3}/{3}
corresponds,
via
the
nat-
ural
identification
just
discussed,
to
an
edge-like
subgroup
of
“Π
1
=
Π
{2}
”
associated
to
the
edge
z
x
∈
Edge(G)
that
appears
in
the
state-
ment
of
Lemma
3.11.
Thus,
we
conclude
immediately
from
[CmbGC],
Proposition
1.2,
(ii),
together
with
the
various
definitions
involved,
∼
that
J
T
=
J
Z
(
←
Z
Π
{1,2,3}/{3}
(T
)).
In
particular,
since
Z
Π
3
(T
)
⊆
N
Π
3
(Z
Π
{1,2,3}/{3}
(T
)),
and
the
surjection
Π
{1,2,3}/{3}
Π
{2,3}/{3}
in-
duces
a
homomorphism
Z
Π
3
(T
)
→
Z
Π
{2,3}/{3}
(J
T
),
one
verifies
easily
that
the
first
inclusion
of
Claim
3.12.B
holds.
The
second
inclusion
of
Claim
3.12.B
follows
from
the
first
inclusion
of
Claim
3.12.B
by
applying
τ
.
This
completes
the
proof
of
Claim
3.12.B,
hence
also
of
Lemma
3.12.
Lemma
3.13
(Preservation
of
verticial
subgroups).
In
the
nota-
tion
of
Lemma
3.11,
let
α
be
an
F-admissible
automorphism
of
Π
E
=
Π
2
,
v
∈
Vert(G).
Write
v
◦
∈
Vert(G
2/1
)
for
the
vertex
of
G
2/1
that
cor-
2/1
responds
to
v
∈
Vert(G)
via
the
bijection
of
Lemma
3.6,
(iv);
α
1
,
α
for
the
automorphisms
of
Π
1
,
Π
2/1
determined
by
α
;
α,
α
1
,
α
2/1
for
the
outomorphisms
of
Π
2
,
Π
1
,
Π
2/1
determined
by
α
,
α
1
,
α
2/1
,
respectively.
Then
the
following
hold:
∼
(i)
Recall
the
edge-like
subgroup
Π
z
x
⊆
Π
1
→
Π
G
associated
to
the
edge
z
x
∈
Edge(G).
Suppose
that
α
1
(Π
z
x
)
=
Π
z
x
.
Suppose,
moreover,
either
that
∼
(a)
the
outomorphism
α
2/1
of
Π
G
2/1
←
Π
2/1
maps
some
cusp-
∼
idal
inertia
subgroup
of
Π
G
2/1
←
Π
2/1
to
a
cuspidal
inertia
∼
subgroup
of
Π
G
2/1
←
Π
2/1
,
or
that
(b)
z
x
∈
Cusp(G).
[For
example,
condition
(a)
holds
if
the
outomorphism
α
2/1
of
∼
Π
G
2/1
←
Π
2/1
is
group-theoretically
cuspidal
—
cf.
[CmbGC],
Definition
1.4,
(iv).]
Then
α
2/1
preserves
the
Π
2/1
-conjugacy
∼
class
of
the
verticial
subgroup
Π
v
new
⊆
Π
2/1
→
Π
G
2/1
associated
to
the
vertex
v
new
∈
Vert(G
2/1
).
If,
moreover,
α
2/1
is
group-
theoretically
cuspidal,
then
the
induced
outomorphism
of
Π
v
new
[cf.
Lemma
3.12,
(i)]
is
itself
group-theoretically
cus-
pidal.
COMBINATORIAL
ANABELIAN
TOPICS
II
71
(ii)
In
the
situation
of
(i),
suppose,
moreover,
that
there
exists
a
∼
∼
verticial
subgroup
Π
v
⊆
Π
G
←
Π
1
of
Π
G
←
Π
1
associated
to
v
∈
Vert(G)
such
that
α
1
preserves
the
Π
1
-conjugacy
class
of
Π
v
.
Then
α
2/1
preserves
the
Π
2/1
-conjugacy
class
of
a
∼
verticial
subgroup
of
Π
G
2/1
←
Π
2/1
associated
to
the
vertex
v
◦
∈
Vert(G
2/1
).
(iii)
In
the
situation
of
(i),
suppose,
moreover,
that
X
log
is
of
type
∼
def
(0,
3)
[which
implies
that
Π
v
=
Π
G
←
Π
1
is
the
unique
verti-
cial
subgroup
of
Π
G
associated
to
v],
and
that
α
1
∈
Out
C
(Π
v
)
cusp
[cf.
Definition
3.4,
(i)].
Then
there
exists
a
geometric
[cf.
∼
∼
Definition
3.4,
(ii)]
outer
isomorphism
Π
v
new
→
Π
v
(=
Π
G
←
Π
1
)
which
satisfies
the
following
condition:
If
either
α
1
∈
Out(Π
1
)
=
Out(Π
v
)
is
contained
in
Out(Π
v
)
Δ
[cf.
Definition
3.4,
(i)]
or
α|
Π
v
new
∈
Out(Π
v
new
)
[cf.
(i);
Lemma
3.12,
(i)]
is
contained
in
Out(Π
v
new
)
Δ
,
then
the
outomorphisms
α|
Π
v
new
,
α
1
of
Π
v
new
,
Π
v
are
compatible
relative
to
the
outer
∼
isomorphism
in
question
Π
v
new
→
Π
v
.
def
Proof.
First,
we
verify
assertions
(i),
(ii).
Write
S
=
Node(G
2/1
)
\
N
(v
new
).
Then
it
follows
immediately
from
the
well-known
local
struc-
ture
of
X
log
in
a
neighborhood
of
the
edge
corresponding
to
z
x
that
if
z
x
∈
Node(G)
(respectively,
z
x
∈
Cusp(G)),
then
the
outer
action
of
Π
z
x
on
Π
(G
2/1
)
S
[cf.
[CbTpI],
Definition
2.8]
obtained
by
conjugating
∼
the
natural
outer
action
Π
z
x
→
Π
1
→
Out(Π
2/1
)
→
Out(Π
G
2/1
)
—
where
the
second
arrow
is
the
outer
action
determined
by
the
exact
sequence
of
profinite
groups
p
Π
2/1
1
−→
Π
2/1
−→
Π
2
−→
Π
1
−→
1
∼
—
by
the
natural
outer
isomorphism
Φ
(G
2/1
)
S
:
Π
(G
2/1
)
S
→
Π
G
2/1
[cf.
[CbTpI],
Definition
2.10]
is
of
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)]
(respectively,
IPSC-type
[cf.
[NodNon],
Definition
2.4,
(i)]).
Thus,
it
follows
immediately
[in
light
of
the
various
assumptions
made
in
the
statement
of
assertion
(i)!]
in
the
case
of
condition
(a)
(respectively,
condition
(b))
from
Theorem
1.9,
(i)
(respectively,
Theorem
1.9,
(ii)),
that
the
outomorphism
α
(G
2/1
)
S
of
Π
(G
2/1
)
S
obtained
by
conjugat-
Φ
(G
∼
2/1
)
S
∼
←
Π
(G
2/1
)
S
is
group-
ing
α
2/1
by
the
composite
Π
2/1
→
Π
G
2/1
theoretically
verticial
[cf.
[CmbGC],
Definition
1.4,
(iv)]
and
group-
theoretically
nodal
[cf.
[NodNon],
Definition
1.12].
On
the
other
hand,
it
follows
immediately
from
condition
(3)
of
[CbTpI],
Proposition
2.9,
∼
(i),
that
the
image
via
Φ
(G
2/1
)
S
:
Π
(G
2/1
)
S
→
Π
G
2/1
of
any
verticial
72
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
subgroup
of
Π
(G
2/1
)
S
associated
to
the
vertex
of
(G
2/1
)
S
correspond-
ing
to
v
new
is
a
verticial
subgroup
of
Π
G
2/1
associated
to
v
new
.
Thus,
since
α
(G
2/1
)
S
is
group-theoretically
verticial,
it
follows
immediately
that
α
2/1
preserves
the
Π
2/1
-conjugacy
class
of
the
verticial
subgroup
∼
Π
v
new
⊆
Π
2/1
→
Π
G
2/1
associated
to
v
new
.
[Here,
we
observe
in
passing
the
following
easily
verified
fact:
a
vertex
of
(G
2/1
)
S
corresponds
to
v
new
if
and
only
if
the
verticial
subgroup
of
Π
(G
2/1
)
S
associated
to
this
p
Π
1\2
∼
vertex
maps,
via
the
composite
Π
(G
2/1
)
S
→
Π
2/1
Π
{2}
,
to
an
abelian
subgroup
of
Π
{2}
.]
If,
moreover,
α
2/1
is
group-theoretically
cuspidal,
then
the
group-theoretic
cuspidality
of
the
resulting
outomorphism
of
Π
v
new
follows
immediately
from
the
group-theoretic
cuspidality
of
α
2/1
and
the
group-theoretic
nodality
of
α
(G
2/1
)
S
.
This
completes
the
proof
of
assertion
(i).
To
verify
assertion
(ii),
let
us
first
observe
that
it
follows
immedi-
ately
from
[CbTpI],
Theorem
A,
(i),
that
—
after
possibly
replacing
α
by
the
composite
of
α
with
an
inner
automorphism
of
Π
2
determined
by
conjugation
by
an
element
of
Π
2/1
—
we
may
assume
without
loss
of
generality
that
if
we
write
α
{2}
for
the
automorphism
of
Π
{2}
deter-
mined
by
α
,
then
α
{2}
(Π
v
)
=
Π
v
—
where,
by
abuse
of
notation,
we
write
Π
v
for
some
fixed
subgroup
of
∼
Π
{2}
whose
image
in
Π
G
←
Π
{2}
is
a
verticial
subgroup
associated
to
v.
∼
Next,
let
us
fix
a
verticial
subgroup
Π
v
◦
⊆
Π
2/1
→
Π
G
2/1
of
Π
G
2/1
as-
sociated
to
the
vertex
v
◦
∈
Vert(G
2/1
)
such
that
the
composite
Π
v
◦
→
p
Π
1\2
∼
Π
2/1
Π
{2}
determines
an
isomorphism
Π
v
◦
→
Π
v
.
Then
let
us
ob-
serve
that
one
verifies
easily
from
condition
(3)
of
[CbTpI],
Proposition
2.9,
(i),
together
with
[NodNon],
Lemma
1.9,
(ii),
that
there
exists
a
unique
vertex
w
◦
∈
Vert((G
2/1
)
S
)
such
that
the
image
Π
w
◦
⊆
Π
2/1
Φ
(G
2/1
)
S
∼
∼
via
the
composite
Π
(G
2/1
)
S
→
Π
G
2/1
←
Π
2/1
of
some
verticial
◦
subgroup
of
Π
(G
2/1
)
S
associated
to
w
contains
the
verticial
subgroup
∼
Π
v
◦
⊆
Π
2/1
→
Π
G
2/1
.
Thus,
it
follows
immediately
from
the
vari-
p
Π
1\2
ous
definitions
involved
that
the
composite
Π
w
◦
→
Π
2/1
Π
{2}
is
an
injective
homomorphism
whose
image
Π
w
⊆
Π
{2}
maps
via
the
∼
Φ
G
S
∼
def
composite
Π
{2}
→
Π
G
←
Π
G
S
—
where
we
write
S
=
Node(G)
\
(Node(G)
∩
{z
x
})
—
to
a
verticial
subgroup
of
Π
G
S
associated
to
a
vertex
w
∈
Vert(G
S
).
Here,
we
note
that
the
vertex
w
may
also
be
characterized
as
the
unique
vertex
of
G
S
such
that
the
image
via
∼
the
natural
outer
isomorphism
Φ
G
S
:
Π
G
S
→
Π
G
of
some
verticial
subgroup
associated
to
w
contains
a
verticial
subgroup
associated
to
COMBINATORIAL
ANABELIAN
TOPICS
II
73
∼
v
∈
Vert(G).
Thus,
we
obtain
an
isomorphism
Π
w
◦
→
Π
w
,
hence
also
∼
an
isomorphism
α
2/1
(Π
w
◦
)
→
α
{2}
(Π
w
).
Next,
let
us
observe
that
since
α
(G
2/1
)
S
is
group-theoretically
ver-
ticial
[cf.
the
argument
given
in
the
proof
of
assertion
(i)],
it
fol-
∼
lows
immediately
that
α
2/1
(Π
w
◦
)
⊆
Π
2/1
→
Π
(G
2/1
)
S
is
a
verticial
subgroup
of
Π
(G
2/1
)
S
that
maps
isomorphically
to
a
verticial
sub-
∼
group
α
{2}
(Π
w
)
⊆
Π
{2}
→
Π
G
S
of
Π
G
S
that
contains
α
{2}
(Π
v
)
=
Π
v
.
On
the
other
hand,
in
light
of
the
unique
characterization
of
w
∼
given
above,
this
implies
that
α
{2}
(Π
w
)
⊆
Π
{2}
→
Π
G
S
is
a
verti-
cial
subgroup
associated
to
w,
and
hence
[as
is
easily
verified]
that
∼
α
2/1
(Π
w
◦
)
⊆
Π
2/1
→
Π
(G
2/1
)
S
is
a
verticial
subgroup
associated
to
w
◦
.
In
particular,
one
may
apply
the
natural
outer
isomorphisms
∼
∼
2/1
(Π
w
◦
);
Π
(G|
H
w
)
Tw
→
α
{2}
(Π
w
)
[cf.
[CbTpI],
Π
((G
2/1
)|
H
w
◦
)
Tw
◦
→
α
Definitions
2.2,
(ii);
2.5,
(ii)]
arising
from
condition
(3)
of
[CbTpI],
Proposition
2.9,
(i);
moreover,
one
verifies
easily
that
the
resulting
∼
outer
isomorphism
Π
((G
2/1
)|
H
w
◦
)
Tw
◦
→
Π
(G|
H
w
)
Tw
[induced
by
the
above
∼
{2}
(Π
w
)]
arises
from
scheme
theory,
hence
isomorphism
α
2/1
(Π
w
◦
)
→
α
is
graphic
[cf.
[CmbGC],
Definition
1.4,
(i)].
Therefore,
we
conclude
∼
that
the
closed
subgroup
α
2/1
(Π
v
◦
)
⊆
(
α
2/1
(Π
w
◦
)
⊆)
Π
2/1
→
Π
G
2/1
is
a
verticial
subgroup
of
Π
G
2/1
associated
to
v
◦
.
This
completes
the
proof
of
assertion
(ii).
Finally,
we
verify
assertion
(iii).
First,
we
recall
from
[CmbCsp],
Corollary
1.14,
(ii),
that
there
exists
an
outer
modular
symmetry
σ
∈
σ
∼
Π
p
2/1
(S
5
⊆)
Out(Π
2
)
such
that
the
composite
Π
v
new
→
Π
2
→
Π
2
Π
1
=
Π
v
determines
a(n)
[necessarily
geometric]
outer
isomorphism
∼
Π
v
new
→
Π
v
.
The
remainder
of
the
proof
of
assertion
(iii)
is
devoted
to
∼
verifying
that
this
outer
isomorphism
Π
v
new
→
Π
v
satisfies
the
condi-
tion
of
assertion
(iii).
First,
suppose
that
α
1
∈
Out(Π
1
)
Δ
.
Then
since
Out
F
(Π
2
)
=
Out
FC
(Π
2
)
=
Out
FCP
(Π
2
)
[cf.
[CmbCsp],
Definition
1.1,
(iv);
Theorem
2.3,
(ii),
(iv),
of
the
present
monograph;
our
assumption
that
X
log
is
of
type
(0,
3)],
it
follows
from
[CmbCsp],
Corollary
1.14,
(i),
together
with
the
injectivity
portion
of
[CmbCsp],
Theorem
A,
(i),
that
α
commutes
with
every
modular
outer
symmetry
on
Π
2
;
in
particular,
α
commutes
with
σ.
Thus,
it
follows
immediately
from
[CmbCsp],
Corol-
∼
lary
1.14,
(iii),
that
the
above
outer
isomorphism
Π
v
new
→
Π
v
satisfies
the
condition
of
assertion
(iii).
def
Next,
suppose
that
α|
Π
v
new
∈
Out(Π
v
new
)
Δ
.
If
we
write
α
σ
=
σ
◦
α
◦
σ
−1
(∈
Out
FC
(Π
2
)
cusp
—
cf.
[CmbCsp],
Corollary
1.14,
(i);
Theo-
rem
2.3,
(ii),
and
Lemma
3.5
of
the
present
monograph)
and
(α
σ
)
1
∈
Out(Π
v
)
for
the
outomorphism
of
Π
v
determined
by
α
σ
,
then
it
fol-
lows
immediately
from
[CmbCsp],
Corollary
1.14,
(iii),
that
the
out-
omorphisms
α|
Π
v
new
,
(α
σ
)
1
of
Π
v
new
,
Π
v
are
compatible
relative
to
the
74
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
outer
isomorphism
Π
v
new
→
Π
v
discussed
above.
Thus,
since
α|
Π
v
new
∈
Out(Π
v
new
)
Δ
,
we
conclude
that
(α
σ
)
1
∈
Out(Π
v
)
Δ
.
In
particular,
[since
Out
F
(Π
2
)
=
Out
FC
(Π
2
)
=
Out
FCP
(Π
2
)
—
cf.
[CmbCsp],
Definition
1.1,
(iv);
Theorem
2.3,
(ii),
(iv),
of
the
present
monograph;
our
assumption
that
X
log
is
of
type
(0,
3)]
it
follows
from
[CmbCsp],
Corollary
1.14,
(i),
together
with
the
injectivity
portion
of
[CmbCsp],
Theorem
A,
(i),
that
α
σ
commutes
with
every
modular
outer
symmetry
on
Π
2
.
Thus,
we
conclude
that
α
σ
commutes
with
σ
−1
,
which
implies
that
α
=
α
σ
.
This
completes
the
proof
of
assertion
(iii).
Lemma
3.14
(Commensurator
of
the
closed
subgroup
arising
from
a
certain
second
log
configuration
space).
Let
i
∈
E,
j
∈
E,
x,
and
z
i,j,x
be
as
in
Lemma
3.6;
let
v
∈
Vert(G
j∈E\{i},x
).
Then,
by
ap-
plying
a
similar
argument
to
the
argument
used
in
[CmbCsp],
Definition
2.1,
(iii),
(vi),
or
[NodNon],
Definition
5.1,
(ix),
(x)
[i.e.,
by
consider-
ing
the
portion
of
the
underlying
scheme
X
E
of
X
E
log
corresponding
to
the
underlying
scheme
(X
v
)
2
of
the
2-nd
log
configuration
space
(X
v
)
log
2
of
the
stable
log
curve
X
v
log
determined
by
G
j∈E\{i},x
|
v
—
cf.
[CbTpI],
Definition
2.1,
(iii)],
one
obtains
a
closed
subgroup
(Π
v
)
2
⊆
Π
E/(E\{i,j})
[which
is
well-defined
up
to
Π
E
-conjugation].
Write
def
(Π
v
)
2/1
=
(Π
v
)
2
∩
Π
E/(E\{i})
⊆
(Π
v
)
2
.
[Thus,
one
verifies
easily
that
there
exists
a
natural
commutative
dia-
gram
1
−−−→
(Π
v
)
2/1
⏐
⏐
−−−→
(Π
v
)
2
⏐
⏐
−−−→
Π
v
⏐
⏐
−−−→
1
p
Π
E/(E\{i})
1
−−−→
Π
E/(E\{i})
−−−→
Π
E/(E\{i,j})
−−−−−−→
Π
(E\{i})/(E\{i,j})
−−−→
1
—
where
we
use
the
notation
Π
v
to
denote
a
verticial
subgroup
of
∼
Π
G
j∈E\{i},x
←
Π
(E\{i})/(E\{i,j})
associated
to
v
∈
Vert(G
j∈E\{i},x
),
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
injective.]
Then
the
following
hold:
(i)
Suppose
that
z
i,j,x
∈
VCN(G
j∈E\{i},x
)
is
contained
in
E(v).
Write
v
◦
∈
Vert(G
i∈E,x
)
for
the
vertex
of
G
i∈E,x
that
corre-
sponds
to
v
∈
Vert(G
j∈E\{i},x
)
via
the
bijections
of
Lemma
3.6,
∼
new
(i),
(iv).
Let
Π
v
◦
,
Π
v
i,j,x
⊆
Π
G
i∈E,x
←
Π
E/(E\{i})
be
verti-
∼
cial
subgroups
of
Π
G
i∈E,x
←
Π
E/(E\{i})
associated
to
the
ver-
new
new
⊆
tices
v
◦
,
v
i,j,x
∈
Vert(G
i∈E,x
),
respectively,
such
that
Π
v
i,j,x
new
(Π
v
)
2/1
,
and,
moreover,
Π
v
◦
∩
Π
v
i,j,x
=
{1}.
Let
us
say
that
two
COMBINATORIAL
ANABELIAN
TOPICS
II
75
Π
E/(E\{i})
-conjugates
Π
γv
◦
,
Π
δv
i,j,x
new
[i.e.,
where
γ,
δ
∈
Π
E/(E\{i})
]
γ
δ
new
are
conjugate-adjacent
if
Π
◦
∩
Π
new
=
{1}.
of
Π
v
◦
,
Π
v
i,j,x
v
v
i,j,x
Let
us
say
that
a
finite
sequence
of
Π
E/(E\{i})
-conjugates
of
Π
v
◦
,
new
is
a
conjugate-chain
if
any
two
adjacent
members
of
Π
v
i,j,x
the
finite
sequence
are
conjugate-adjacent.
Let
us
say
that
a
subgroup
of
Π
E/(E\{i})
is
conjugate-tempered
if
it
appears
as
the
first
member
of
a
conjugate-chain
whose
final
mem-
new
.
Then
(Π
v
)
2/1
is
equal
to
the
subgroup
ber
is
equal
to
Π
v
i,j,x
of
Π
E/(E\{i})
topologically
generated
by
the
conjugate-tempered
subgroups
and
the
elements
δ
∈
Π
E/(E\{i})
such
that
Π
δv
i,j,x
new
is
conjugate-tempered.
(ii)
If
N
Π
E\{i}
(Π
v
)
=
C
Π
E\{i}
(Π
v
),
then
N
Π
E
((Π
v
)
2
)
=
C
Π
E
((Π
v
)
2
).
(iii)
If
C
Π
E\{i}
(Π
v
)
=
Π
v
×
Z
Π
E\{i}
(Π
v
),
then
C
Π
E
((Π
v
)
2
)
=
(Π
v
)
2
×
Z
Π
E
((Π
v
)
2
).
(iv)
Suppose
that
v
is
of
type
(0,
3),
i.e.,
that
Π
v
is
an
(E
\
{i})-tripod
of
Π
n
[cf.
Definition
3.3,
(i)].
Then
it
holds
that
C
Π
E
((Π
v
)
2
)
=
(Π
v
)
2
×
Z
Π
E
((Π
v
)
2
).
Thus,
if
an
outomorphism
α
of
Π
E
preserves
the
Π
E
-conjugacy
class
of
(Π
v
)
2
,
then
one
may
define
α|
(Π
v
)
2
∈
Out((Π
v
)
2
)
[cf.
Lemma
3.10,
(i)].
Proof.
First,
we
verify
assertion
(i).
We
begin
by
observing
that
it
follows
immediately
from
[NodNon],
Lemma
1.9,
(ii),
together
with
new
⊆
Π
E/(E\{i})
[cf.
the
commensurable
terminality
of
Π
v
i,j,x
[CmbGC],
Proposition
1.2,
(ii)],
that
the
subgroup
described
in
the
final
portion
of
the
statement
of
assertion
(i)
is
contained
in
(Π
v
)
2/1
.
If
#(N
(v
◦
)
∩
new
N
(v
i,j,x
))
=
1,
then
assertion
(i)
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
[CmbCsp],
Propo-
sition
1.5,
(iii),
together
with
the
various
definitions
involved
[cf.
also
[NodNon],
Lemma
1.9,
(ii)].
Thus,
we
may
assume
without
loss
of
new
))
=
2.
generality
that
#(N
(v
◦
)
∩
N
(v
i,j,x
Write
new
•
e
1
∈
N
(v
◦
)
∩
N
(v
i,j,x
)
for
the
[uniquely
determined
—
cf.
new
(
=
{1})
[NodNon],
Lemma
1.5]
node
such
that
Π
v
◦
∩
Π
v
i,j,x
is
a
nodal
subgroup
associated
to
e
1
[cf.
[NodNon],
Lemma
1.9,
(i)];
new
•
e
2
for
the
unique
element
of
N
(v
◦
)
∩
N
(v
i,j,x
)
such
that
e
2
=
e
1
◦
new
[so
N
(v
)
∩
N
(v
i,j,x
)
=
{e
1
,
e
2
}];
•
H
for
the
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
of
the
underlying
semi-graph
of
G
i∈E,x
whose
set
of
new
vertices
=
{v
◦
,
v
i,j,x
};
def
•
S
=
Node(G
i∈E,x
|
H
)
\
{e
1
,
e
2
}
[cf.
[CbTpI],
Definition
2.2,
(ii)];
76
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
def
•
H
=
(G
i∈E,x
|
H
)
S
[which
is
well-defined
since,
as
is
easily
ver-
ified,
S
is
not
of
separating
type
as
a
subset
of
Node(G
i∈E,x
|
H
)
—
cf.
[CbTpI],
Definition
2.5,
(i),
(ii)].
Then
it
follows
immediately
from
the
construction
of
H
that
H
{e
1
}
[cf.
[CbTpI],
Definition
2.8],
where
we
observe
that
one
verifies
eas-
ily
that
the
node
e
1
of
G
i∈E,x
may
be
regarded
as
a
node
of
H,
is
cyclically
primitive
[cf.
[CbTpI],
Definition
4.1].
Moreover,
it
follows
immediately
from
[NodNon],
Lemma
1.9,
(ii),
together
with
the
vari-
∼
ous
definitions
involved,
that
(Π
v
)
2/1
⊆
Π
E/(E\{i})
→
Π
G
i∈E,x
may
be
characterized
uniquely
as
the
closed
subgroup
of
Π
G
i∈E,x
that
contains
new
⊆
Π
G
Π
v
i,j,x
and,
moreover,
belongs
to
the
Π
G
i∈E,x
-conjugacy
class
i∈E,x
of
closed
subgroups
of
Π
G
i∈E,x
obtained
by
forming
the
image
of
the
composite
of
outer
homomorphisms
Π
H
{e
1
}
Φ
H
{e
}
1
∼
→
Π
H
→
Π
G
i∈E,x
[cf.
[CbTpI],
Definition
2.10]
—
where
the
second
arrow
is
the
outer
in-
jection
discussed
in
[CbTpI],
Proposition
2.11.
In
particular,
it
follows
from
the
commensurable
terminality
of
(Π
v
)
2/1
in
Π
G
i∈E,x
[cf.
[CmbGC],
Proposition
1.2,
(ii)]
that
this
characterization
of
(Π
v
)
2/1
determines
an
∼
outer
isomorphism
Π
H
{e
1
}
→
(Π
v
)
2/1
.
On
the
other
hand,
it
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
[CmbCsp],
Proposition
1.5,
(iii),
together
with
the
various
definitions
involved
[cf.
also
[NodNon],
Lemma
1.9,
(ii)],
that
the
image
of
the
closed
subgroup
of
(Π
v
)
2/1
topo-
∼
new
via
the
inverse
(Π
v
)
2/1
→
Π
H
logically
generated
by
Π
v
◦
and
Π
v
i,j,x
{e
1
}
of
this
outer
isomorphism
is
a
verticial
subgroup
of
Π
H
{e
1
}
associated
to
the
unique
vertex
of
H
{e
1
}
.
Thus,
since
H
{e
1
}
is
cyclically
primi-
tive,
assertion
(i)
follows
immediately
from
[CmbGC],
Proposition
1.2,
(ii);
[NodNon],
Lemma
1.9,
(ii),
together
with
the
description
of
the
structure
of
a
certain
tempered
covering
of
H
{e
1
}
given
in
[CbTpI],
Lemma
4.3.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Since
(Π
v
)
2/1
=
(Π
v
)
2
∩
Π
E/(E\{i})
is
commensurably
terminal
in
Π
E/(E\{i})
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
assertion
(ii)
follows
immediately
from
Lemma
3.9,
(iv).
This
completes
the
proof
of
assertion
(ii).
Next,
we
verify
assertion
(iii).
First,
let
us
observe
that
if
E(v)
=
∅,
then
one
verifies
immediately
that
the
vertical
arrows
of
the
commutative
diagram
in
the
statement
of
Lemma
3.14
are
isomorphisms,
and
hence
that
assertion
(iii)
holds.
Thus,
we
may
assume
that
E(v)
=
∅.
Next,
let
us
observe
that
it
follows
from
assertion
(ii)
that
N
Π
E
((Π
v
)
2
)
=
C
Π
E
((Π
v
)
2
).
Thus,
in
light
of
the
slimness
of
(Π
v
)
2
[cf.
[MzTa],
Proposition
2.2,
(ii)],
to
verify
assertion
(iii),
it
suffices
to
verify
that
the
natural
outer
action
of
N
Π
E
((Π
v
)
2
)
on
(Π
v
)
2
is
trivial.
On
the
other
hand,
since
[one
verifies
easily
that]
COMBINATORIAL
ANABELIAN
TOPICS
II
77
the
natural
outer
action
N
Π
E
((Π
v
)
2
)
→
Out((Π
v
)
2
)
factors
through
Out
F
((Π
v
)
2
)
⊆
Out((Π
v
)
2
),
it
follows
from
the
injectivity
portion
of
Theorem
2.3,
(i)
[cf.
our
assumption
that
E(v)
=
∅],
that
to
verify
the
triviality
in
question,
it
suffices
to
verify
that
the
natural
outer
action
of
N
Π
E
((Π
v
)
2
)
on
Π
v
is
trivial.
But
this
follows
from
the
equality
C
Π
E\{i}
(Π
v
)
=
Π
v
×
Z
Π
E\{i}
(Π
v
).
This
completes
the
proof
of
assertion
(iii).
Assertion
(iv)
follows
immediately
from
assertion
(iii),
together
with
Lemma
3.12,
(i).
This
completes
the
proof
of
Lemma
3.14.
Lemma
3.15
(Preservation
of
various
subgroups
of
geomet-
ric
origin).
In
the
notation
of
Lemma
3.14,
let
α
be
an
F-admissible
automorphism
of
Π
E
.
Write
α
E\{i}
,
α
E/(E\{i})
for
the
automorphisms
;
α,
α
E\{i}
,
α
E/(E\{i})
for
the
outo-
of
Π
E\{i}
,
Π
E/(E\{i})
determined
by
α
,
α
E\{i}
,
α
E/(E\{i})
,
morphisms
of
Π
E
,
Π
E\{i}
,
Π
E/(E\{i})
determined
by
α
respectively.
Suppose
that
there
exist
an
edge
e
∈
Edge(G
j∈E\{i},x
)
of
G
j∈E\{i},x
that
belongs
to
E(v)
⊆
Edge(G
j∈E\{i},x
)
and
a
pair
Π
e
⊆
∼
Π
v
⊆
Π
G
j∈E\{i},x
←
Π
(E\{i})/(E\{i,j})
of
VCN-subgroups
associated
to
e
∈
Edge(G
j∈E\{i},x
),
v
∈
Vert(G
j∈E\{i},x
),
respectively,
such
that
α
E\{i}
(Π
e
)
=
Π
e
⊆
α
E\{i}
(Π
v
)
=
Π
v
.
Suppose,
moreover,
either
that
∼
(a)
the
outomorphism
α
E/(E\{i})
of
Π
G
i∈E,x
←
Π
E/(E\{i})
maps
some
∼
cuspidal
inertia
subgroup
of
Π
G
i∈E,x
←
Π
E/(E\{i})
to
a
cuspidal
∼
inertia
subgroup
of
Π
G
i∈E,x
←
Π
E/(E\{i})
,
or
that
(b)
e
∈
Cusp(G
j∈E\{i},x
).
[For
example,
condition
(a)
holds
if
the
outomorphism
α
E/(E\{i})
of
∼
Π
G
i∈E,x
←
Π
E/(E\{i})
is
group-theoretically
cuspidal
—
cf.
[CmbGC],
Definition
1.4,
(iv).]
Write
T
⊆
Π
E
for
the
E-tripod
of
Π
n
[cf.
Def-
inition
3.3,
(i)]
arising
from
e
∈
Edge(G
j∈E\{i},x
)
[cf.
Definition
3.7,
(i)].
Then
the
following
hold:
(i)
The
outomorphism
α
preserves
the
Π
E
-conjugacy
classes
of
T
,
(Π
v
)
2
⊆
Π
E
.
If,
moreover,
the
outomorphism
α
E/(E\{i})
∼
of
Π
G
i∈E,x
←
Π
E/(E\{i})
is
group-theoretically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(iv)],
then
the
outomorphism
α|
T
[cf.
Lemma
3.12,
(i)]
of
T
is
contained
in
Out
C
(T
)
cusp
[cf.
Definition
3.4,
(i)].
(ii)
Suppose,
moreover,
that
v
is
of
type
(0,
3)
—
i.e.,
that
Π
v
is
an
(E\{i})-tripod
of
Π
n
—
and
that
α
E\{i}
|
Π
v
∈
Out
C
(Π
v
)
cusp
[cf.
Lemma
3.12,
(i)].
Then
there
exists
a
geometric
[cf.
Def-
∼
inition
3.4,
(ii)]
outer
isomorphism
T
→
Π
v
which
satisfies
the
following
condition:
78
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
If
either
α|
T
∈
Out(T
)
Δ
[cf.
(i)]
or
α
E\{i}
|
Π
v
∈
Out(Π
v
)
Δ
,
then
the
outomorphisms
α|
T
,
α
E\{i}
|
Π
v
of
T
,
Π
v
are
compatible
relative
to
the
outer
isomor-
∼
phism
in
question
T
→
Π
v
.
If,
moreover,
Π
v
is
(E
\
{i})-strict
[cf.
Definition
3.3,
(iii)],
then
the
following
hold:
(1)
If
#(E\{i})
=
1
[i.e.,
Π
v
satisfies
condition
(1)
of
Lemma
3.8,
(ii)],
then
T
is
E-strict
[i.e.,
T
satisfies
one
of
the
two
conditions
(2
C
),
(2
N
)
of
Lemma
3.8,
(ii)].
(2)
If
#(E
\
{i})
=
2
[i.e.,
Π
v
satisfies
one
of
the
two
con-
ditions
(2
C
),
(2
N
)
of
Lemma
3.8,
(ii)],
and
the
edge
e
∈
Edge(G
j∈E\{i},x
)
is
the
unique
diagonal
cusp
of
G
j∈E\{i},x
[cf.
Lemma
3.2,
(ii)],
then
T
is
E-strict
[i.e.,
T
satisfies
condition
(3)
of
Lemma
3.8,
(ii)],
hence
also
central
[cf.
Definition
3.7,
(ii)].
Proof.
First,
let
us
observe
that
one
verifies
easily
—
by
replacing
x
by
a
suitable
k-valued
geometric
point
of
X
n
(k)
that
lifts
x
E\{i,j}
∈
X
E\{i,j}
(k)
[note
that
this
does
not
affect
“G
j∈E\{i},x
”!]
—
that,
to
verify
Lemma
3.15,
we
may
assume
without
loss
of
generality
that
z
i,j,x
=
e
∈
Edge(G
j∈E\{i},x
).
Now
we
verify
assertion
(i).
First,
let
us
observe
that
one
verifies
eas-
log
log
ily
—
by
replacing
X
E
log
by
the
base-change
of
p
log
E\{i,j}
:
X
E
→
X
E\{i,j}
log
by
a
suitable
morphism
of
log
schemes
(Spec
k)
log
→
X
E\{i,j}
that
lies
over
x
E\{i,j}
∈
X
E\{i,j}
(k)
[cf.
Definition
3.1,
(i)]
—
that,
to
verify
asser-
tion
(i),
we
may
assume
without
loss
of
generality
that
#E
=
2.
Then
it
follows
immediately
from
Lemma
3.13,
(i),
that
α
E/(E\{i})
preserves
the
new
)
⊆
Π
E/(E\{i})
.
Moreover,
it
fol-
Π
E/(E\{i})
-conjugacy
class
of
T
(=
Π
v
i,j,x
lows
immediately
from
Lemma
3.13,
(i),
(ii),
together
with
Lemma
3.14,
(i),
that
α
E/(E\{i})
preserves
the
Π
E/(E\{i})
-conjugacy
classes
of
the
normally
terminal
closed
subgroups
Π
v
◦
⊆
(Π
v
)
2/1
⊆
Π
E/(E\{i})
[cf.
[CmbGC],
Proposition
1.2,
(ii)].
In
particular,
since
α
E\{i}
(Π
v
)
=
Π
v
,
∼
out
by
considering
the
natural
isomorphism
(Π
v
)
2
→
(Π
v
)
2/1
Π
v
[cf.
the
upper
exact
sequence
of
the
commutative
diagram
in
the
state-
ment
of
Lemma
3.14;
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0],
we
conclude
that
α
E
preserves
the
Π
E
-conjugacy
class
of
(Π
v
)
2
⊆
Π
E
.
∼
Next,
suppose
that
the
outomorphism
α
E/(E\{i})
of
Π
G
i∈E,x
←
Π
E/(E\{i})
is
group-theoretically
cuspidal.
Then
it
follows
from
Lemma
3.13,
(i),
that
α|
T
∈
Out
C
(T
).
Moreover,
since
α
E/(E\{i})
is
group-theoretically
cuspidal,
it
follows
immediately
from
Lemma
3.2,
(iv),
that
α
E/(E\{i})
fixes
the
Π
E/(E\{i})
-conjugacy
class
of
cuspidal
inertia
subgroups
asso-
new
)
(
c
diag
ciated
to
each
element
∈
C(v
i,j,x
i,j,x
).
Thus,
to
verify
that
α|
T
∈
COMBINATORIAL
ANABELIAN
TOPICS
II
79
Out
C
(T
)
cusp
,
it
suffices
to
verify
that
α
E/(E\{i})
fixes
the
Π
E/(E\{i})
-
∼
conjugacy
class
of
nodal
subgroups
of
Π
G
i∈E,x
←
Π
E/(E\{i})
associated
to
new
new
each
element
of
N
(v
i,j,x
)∩N
(v
◦
).
To
this
end,
let
e
◦
∈
N
(v
i,j,x
)∩N
(v
◦
)
∼
and
Π
e
◦
⊆
Π
G
i∈E,x
←
Π
E/(E\{i})
a
nodal
subgroup
associated
to
the
node
e
◦
such
that
Π
e
◦
⊆
Π
v
◦
.
Now
let
us
observe
that
one
verifies
∼
easily
that
the
closed
subgroups
Π
e
◦
⊆
Π
v
◦
⊆
Π
G
i∈E,x
←
Π
E/(E\{i})
∼
map
bijectively
onto
VCN-subgroups
of
Π
G
i∈E\{j}
,x
←
Π
(E\{j})/(E\{i,j})
associated,
respectively,
to
the
edge
and
vertex
of
G
i∈E\{j},x
that
corre-
spond,
via
the
bijections
of
Lemma
3.6,
(i),
to
e,
v
∈
VCN(G
j∈E\{i},x
).
In
particular,
if
β
is
the
composite
of
α
with
some
Π
E/(E\{i})
-inner
v
◦
)
=
Π
v
◦
[cf.
the
preceding
paragraph],
automorphism
such
that
β(Π
then
it
follows
immediately
from
our
assumption
that
α
E\{i}
(Π
e
)
=
Π
e
⊆
α
E\{i}
(Π
v
)
=
Π
v
,
together
with
[CbTpI],
Theorem
A,
(i),
and
[CmbGC],
Proposition
1.2,
(ii),
that
the
automorphism
of
Π
v
◦
deter-
mined
by
β
preserves
the
Π
v
◦
-conjugacy
class
of
Π
e
◦
.
Thus,
α
E/(E\{i})
fixes
the
Π
E/(E\{i})
-conjugacy
class
of
Π
e
◦
,
as
desired.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
Since
v
is
of
type
(0,
3),
it
follows
from
assertion
(i),
together
with
Lemma
3.14,
(iv),
that
one
may
de-
fine
α|
(Π
v
)
2
∈
Out((Π
v
)
2
).
Thus,
by
applying
Lemma
3.13,
(iii),
to
α|
(Π
v
)
2
∈
Out((Π
v
)
2
),
one
verifies
easily
that
the
first
portion
of
asser-
tion
(ii)
holds.
The
final
portion
of
assertion
(ii)
follows
immediately
from
the
descriptions
given
in
the
four
conditions
of
Lemma
3.8,
(ii),
together
with
the
various
definitions
involved.
This
completes
the
proof
of
assertion
(ii).
Theorem
3.16
(Outomorphisms
preserving
tripods).
In
the
no-
tation
of
the
beginning
of
the
present
§3,
let
E
⊆
{1,
·
·
·
,
n}
and
T
⊆
Π
E
an
E-tripod
of
Π
n
[cf.
Definition
3.3,
(i)].
Let
us
write
Out
F
(Π
n
)[T
]
⊆
Out
F
(Π
n
)
for
the
[closed]
subgroup
of
Out
F
(Π
n
)
[cf.
[CmbCsp],
Definition
1.1,
(ii)]
consisting
of
F-admissible
outomorphisms
α
of
Π
n
such
that
the
outomorphism
of
Π
E
determined
by
α
preserves
the
Π
E
-conjugacy
class
of
T
⊆
Π
E
.
Then
the
following
hold:
(i)
It
holds
that
C
Π
E
(T
)
=
T
×
Z
Π
E
(T
).
Thus,
by
applying
Lemma
3.10,
(i),
to
outomorphisms
of
Π
E
determined
by
elements
of
Out
F
(Π
n
)[T
],
one
obtains
a
natural
homomorphism
T
T
:
Out
F
(Π
n
)[T
]
−→
Out(T
).
80
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Let
us
write
Out
F
(Π
n
)[T
:
{C}],
Out
F
(Π
n
)[T
:
{|C|}],
Out
F
(Π
n
)[T
:
{Δ}],
Out
F
(Π
n
)[T
:
{+}]
⊆
Out
F
(Π
n
)[T
]
for
the
[closed]
subgroups
of
Out
F
(Π
n
)[T
]
obtained
by
forming
the
respective
inverse
images
via
T
T
of
the
closed
subgroups
Out
C
(T
),
Out
C
(T
)
cusp
,
Out(T
)
Δ
,
Out(T
)
+
⊆
Out(T
)
[cf.
Def-
inition
3.4,
(i)].
For
each
subset
S
⊆
{C,
|C|,
Δ,
+},
let
us
write
def
Out
F
(Π
n
)[T
:
S]
=
Out
F
(Π
n
)[T
:
{}]
⊆
Out
F
(Π
n
)[T
]
;
∈S
def
Out
FC
(Π
n
)[T
:
S]
=
Out
F
(Π
n
)[T
:
S]
∩
Out
FC
(Π
n
)
⊆
Out
FC
(Π
n
)
[cf.
[CmbCsp],
Definition
1.1,
(ii)].
Suppose,
moreover,
that
we
are
given
an
element
σ
∈
S
n
⊆
Out(Π
n
)
[cf.
the
discussion
at
the
beginning
of
the
present
§3]
and
a
lifting
σ
∈
Aut(Π
n
)
of
σ
∈
S
n
⊆
Out(Π
n
).
Write
T
σ
⊆
Π
σ(E)
∼
for
the
image
of
T
⊆
Π
E
by
the
isomorphism
Π
E
→
Π
σ(E)
determined
by
σ
∈
Aut(Π
n
)
[which
thus
implies
that
T
σ
⊆
Π
σ(E)
is
a
σ(E)-tripod
of
Π
n
—
cf.
Remark
3.7.1]
and
]
=
Out
F
(Π
n
)[T
]
∩
Out
F
(Π
n
)[T
σ
]
⊆
Out
F
(Π
n
),
Out
F
(Π
n
)[T,
σ
def
def
Out
FC
(Π
n
)[T,
σ
]
=
Out
F
(Π
n
)[T,
σ
]∩Out
FC
(Π
n
)
⊆
Out
FC
(Π
n
).
∼
Then
the
resulting
isomorphism
T
→
T
σ
is
geometric
[cf.
Definition
3.4,
(ii)].
Moreover,
we
have
a
commutative
dia-
gram
Out
F
(Π
n
)[T,
σ
]
⏐
⏐
T
T
Out
F
(Π
n
)[T,
σ
]
⏐
⏐
T
T
σ
∼
Out(T
)
−−−→
Out(T
σ
)
—
where
the
upper
horizontal
equality
is
an
equality
of
sub-
groups
of
the
group
Out
F
(Π
n
),
and
the
lower
horizontal
arrow
is
the
isomorphism
obtained
by
conjugating
by
the
above
geo-
∼
∈
Aut(Π
n
)].
metric
isomorphism
T
→
T
σ
[i.e.,
induced
by
σ
Finally,
the
equalities
]
=
Out
FC
(Π
n
)[T
]
=
Out
FC
(Π
n
)[T
σ
]
Out
FC
(Π
n
)[T,
σ
hold;
if,
moreover,
one
of
the
following
conditions
is
satisfied,
then
the
equalities
Out
F
(Π
n
)[T,
σ
]
=
Out
F
(Π
n
)[T
]
=
Out
F
(Π
n
)[T
σ
]
hold:
COMBINATORIAL
ANABELIAN
TOPICS
II
81
(i-1)
(r,
n)
=
(0,
2).
(i-2)
T
is
E-strict
[cf.
Definition
3.3,
(iii)].
(ii)
It
holds
that
Out
F
(Π
n
)[T
:
{C,
Δ}]
=
Out
F
(Π
n
)[T
:
{|C|,
Δ}]
.
(iii)
Suppose
that
T
is
1-descendable
[cf.
Definition
3.3,
(iv)].
Then
it
holds
that
Out
FC
(Π
n
)[T
:
{|C|}]
=
Out
FC
(Π
n
)[T
:
{|C|,
+}]
.
If,
moreover,
one
of
the
following
conditions
is
satisfied,
then
it
holds
that
Out
F
(Π
n
)[T
:
{|C|}]
=
Out
F
(Π
n
)[T
:
{|C|,
+}]
:
(iii-1)
T
is
2-descendable
[cf.
Definition
3.3,
(iv)].
(iii-2)
There
exists
a
subset
E
⊆
E
such
that:
(iii-2-a)
E
=
{1,
·
·
·
,
n};
(iii-2-b)
the
image
p
Π
E/E
(T
)
⊆
Π
E
is
a
cusp-supporting
E
-tripod
of
Π
n
[cf.
Definition
3.3,
(i)].
(iv)
Let
i,
j
∈
E
be
two
distinct
elements
of
E;
e
∈
Edge(G
j∈E\{i},x
)
[cf.
Definition
3.1,
(iii)];
α
∈
Out
F
(Π
n
).
Suppose
that
T
arises
from
e
∈
Edge(G
j∈E\{i},x
)
[cf.
Definition
3.7,
(i)],
and
that
the
outomorphism
of
Π
E\{i}
determined
by
α
preserves
the
Π
E\{i}
-conjugacy
class
of
an
edge-like
subgroup
of
Π
E\{i}
associated
to
e
∈
Edge(G
j∈E\{i},x
)
[cf.
Definition
3.1,
(iv)].
Suppose,
moreover,
that
one
of
the
following
conditions
is
sat-
isfied:
(iv-1)
α
∈
Out
FC
(Π
n
).
(iv-2)
#E
≤
n
−
1.
(iv-3)
e
∈
Cusp(G
j∈E\{i},x
).
Then
α
∈
Out
F
(Π
n
)[T
].
Suppose,
further,
that
either
condition
(iv-1)
or
condition
(iv-2)
is
satisfied.
Then
α
∈
Out
F
(Π
n
)[T
:
{C}];
if,
in
addition,
condition
(iv-3)
is
satisfied,
then
α
∈
Out
F
(Π
n
)[T
:
{|C|}].
(v)
Suppose
that
T
is
central
[cf.
Definition
3.7,
(ii)].
If
n
≥
4
[i.e.,
T
is
1-descendable],
then
it
holds
that
Out
F
(Π
n
)
=
Out
FC
(Π
n
)[T
:
{|C|,
Δ,
+}]
.
If
n
=
3
[i.e.,
T
is
not
1-descendable],
then
it
holds
that
Out
FC
(Π
n
)
=
Out
FC
(Π
n
)[T
:
{|C|,
Δ}]
⊆
Out
F
(Π
n
)
=
Out
F
(Π
n
)[T
:
{Δ}]
;
82
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
if,
moreover,
r
=
0,
then
Out
F
(Π
n
)
=
Out
FC
(Π
n
)[T
:
{|C|,
Δ,
+}]
.
Proof.
We
begin
the
proof
of
Theorem
3.16
with
the
following
claim:
Claim
3.16.A:
Let
E
⊆
E
be
a
subset
such
that
the
image
T
E
of
T
via
p
Π
E/E
:
Π
E
Π
E
is
an
E
-tripod.
Thus,
one
verifies
easily
that
one
obtains
a(n)
[neces-
∼
sarily
geometric]
outer
isomorphism
T
→
T
E
[induced
F
by
p
Π
E/E
].
Then
we
have
an
inclusion
Out
(Π
n
)[T
]
⊆
Out
F
(Π
n
)[T
E
],
and,
moreover,
the
diagram
Out
F
(Π
n
)[T
]
⊆
T
T
↓
Out(T
)
Out
F
(Π
n
)[T
E
]
↓
T
T
E
∼
−→
Out(T
E
)
—
where
the
lower
horizontal
arrow
is
the
isomorphism
∼
determined
by
the
isomorphism
T
→
T
E
induced
by
p
Π
E/E
—
commutes.
Indeed,
this
follows
immediately
from
the
various
definitions
involved.
This
completes
the
proof
of
Claim
3.16.A.
Next,
we
verify
assertion
(i).
The
equality
C
Π
E
(T
)
=
T
×
Z
Π
E
(T
)
of
the
first
display
in
assertion
(i)
follows
from
Lemma
3.12,
(i).
Moreover,
∼
the
geometricity
of
the
isomorphism
T
→
T
σ
follows
immediately
from
the
various
definitions
involved.
Next,
let
us
observe
that
if
(r,
n)
=
(0,
2),
then
the
commutativity
of
the
displayed
diagram
in
assertion
(i)
and
the
equalities
]
=
Out
F
(Π
n
)[T
]
=
Out
F
(Π
n
)[T
σ
]
Out
F
(Π
n
)[T,
σ
in
assertion
(i)
may
be
easily
derived
from
the
fact
that
the
closed
subgroup
Out
F
(Π
n
)
⊆
Out(Π
n
)
centralizes
the
closed
subgroup
S
n
⊆
Out
F
(Π
n
)
[cf.
Theorem
2.3,
(iv)].
Moreover,
the
equalities
Out
FC
(Π
n
)[T,
σ
]
=
Out
FC
(Π
n
)[T
]
=
Out
FC
(Π
n
)[T
σ
]
in
assertion
(i)
may
be
easily
derived
from
the
fact
that
the
closed
subgroup
Out
FC
(Π
n
)
⊆
Out(Π
n
)
centralizes
the
closed
subgroup
S
n
⊆
Out
F
(Π
n
)
[cf.
[NodNon],
Theorem
B].
Next,
let
us
observe
that
if
T
is
E
-strict
for
some
subset
E
⊆
E
of
cardinality
one,
then
the
commutativity
of
the
displayed
diagram
in
assertion
(i)
follows
immediately
from
Claim
3.16.A
and
[CbTpI],
The-
orem
A,
(i).
Thus,
it
follows
from
Lemma
3.8,
(ii),
that,
to
complete
the
verification
of
assertion
(i),
it
suffices
to
verify,
under
the
assumption
that
σ
=
id,
COMBINATORIAL
ANABELIAN
TOPICS
II
83
(a)
the
commutativity
of
the
displayed
diagram
in
assertion
(i)
in
the
case
where
(r,
n)
=
(0,
2),
and
T
is
{1,
2}-strict,
and
(b)
the
equalities
Out
F
(Π
n
)[T,
σ
]
=
Out
F
(Π
n
)[T
]
=
Out
F
(Π
n
)[T
σ
]
in
assertion
(i)
in
the
case
where
(r,
n)
=
(0,
2),
and
T
is
{1,
2}-
strict.
In
particular,
to
verify
assertion
(i),
we
may
assume
without
loss
of
generality
[cf.
conditions
(2
C
)
and
(2
N
)
of
Lemma
3.8,
(ii)]
that
we
are
in
the
situation
of
Lemma
3.11
in
the
case
where
we
take
the
“n”,
“E”
of
Lemma
3.11
to
be
2,
{1,
2},
respectively.
Moreover,
it
follows
immediately
from
Lemma
3.8,
(ii),
that
the
Π
n
-conjugacy
classes
of
T
,
T
σ
coincide
with
the
Π
n
-conjugacy
classes
of
the
closed
subgroups
new
,
Π
v
new
of
Π
n
that
appear
in
the
statement
of
Lemma
3.11,
re-
Π
v
2/1
1\2
spectively.
Then
the
above
equalities
in
(b)
follows
immediately
from
Lemma
3.11,
(x).
Moreover,
it
follows
from
Lemma
3.11,
(viii),
(ix),
that
the
composites
T
→
C
Π
n
(T
)
C
Π
n
(T
)/Z(C
Π
n
(T
)),
T
σ
→
C
Π
n
(T
σ
)
C
Π
n
(T
σ
)/Z(C
Π
n
(T
σ
))
are
isomorphisms.
Thus,
the
commutativity
in
(a)
follows
immediately
from
Lemma
3.11,
(x).
This
completes
the
proof
of
assertion
(i).
As-
sertion
(ii)
follows
from
Lemma
3.5.
Next,
we
verify
assertion
(iii).
First,
to
verify
the
first
displayed
equality
of
assertion
(iii),
let
us
observe
that
since
T
is
1-descendable,
there
exists
a
subset
E
⊆
E
such
that
the
image
of
T
⊆
Π
E
via
p
Π
E/E
:
Π
E
Π
E
is
an
E
-tripod,
and,
moreover,
#E
≤
n
−
1.
Thus,
it
follows
immediately
from
Claim
3.16.A,
together
with
Remark
3.4.1
—
by
replacing
T
,
E,
by
p
Π
E/E
(T
),
E
,
respectively
—
that,
to
verify
the
first
displayed
equality
of
assertion
(iii),
we
may
assume
without
loss
of
generality
that
E
=
{1,
·
·
·
,
n}.
Then
the
first
displayed
equality
of
assertion
(iii)
follows
immediately
from
Lemma
3.14,
(iv);
the
portion
of
Lemma
3.15,
(i)
[where
we
observe
that
the
“T
”
of
Lemma
3.15
differs
from
the
T
of
the
present
discussion!],
concerning
“(Π
v
)
2
”
[cf.
condition
(a)
of
Lemma
3.15].
This
completes
the
proof
of
the
first
displayed
equality
of
assertion
(iii).
Next,
suppose
that
condition
(iii-1)
is
satisfied;
thus,
there
exists
a
subset
E
⊆
E
such
that
the
image
p
Π
E/E
(T
)
⊆
Π
E
is
an
E
-tripod,
and,
moreover,
#E
≤
n
−
2.
Then
—
by
replacing
T
,
E
by
p
Π
E/E
(T
),
E
,
respectively
[and
applying
Claim
3.16.A]
—
we
may
assume
without
loss
of
generality
that
#E
≤
n
−
2.
Thus,
by
applying
[CbTpI],
Theo-
rem
A,
(ii),
we
conclude
that
the
second
displayed
equality
of
assertion
(iii)
follows
immediately
from
the
first
displayed
equality
of
assertion
(iii).
84
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Next,
suppose
that
condition
(iii-2)
is
satisfied.
Then
—
by
re-
placing
T
,
E
by
the
p
Π
E/E
(T
),
E
in
condition
(iii-2)
[and
applying
Claim
3.16.A]
—
we
may
assume
without
loss
of
generality
that
E
=
{1,
·
·
·
,
n},
and,
moreover,
that
T
is
a
cusp-supporting
E-tripod.
Then
it
follows
immediately
from
Lemma
3.14,
(iv);
the
portion
of
Lemma
3.15,
(i),
concerning
(Π
v
)
2
[cf.
condition
(b)
of
Lemma
3.15],
that
the
second
displayed
equality
of
assertion
(iii)
holds.
This
completes
the
proof
of
assertion
(iii).
Next,
we
verify
assertion
(iv).
If
either
condition
(iv-1)
or
condi-
tion
(iv-3)
is
satisfied,
then
one
reduces
immediately
to
the
case
where
n
=
2,
in
which
case
it
follows
immediately
from
Lemma
3.13,
(i),
that
α
∈
Out
F
(Π
n
)[T
].
If
condition
(iv-1)
is
satisfied,
then
one
reduces
im-
mediately
to
the
case
where
n
=
2,
in
which
case
it
follows
immediately
from
Lemma
3.13,
(i),
that
α
∈
Out
F
(Π
n
)[T
:
{C}].
If
both
condition
(iv-1)
and
condition
(iv-3)
are
satisfied,
then
—
by
applying
a
suit-
able
specialization
isomorphism
[cf.
the
discussion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1]
—
one
reduces
imme-
diately
to
the
case
where
n
=
2
and
Node(G)
=
∅,
in
which
case
it
fol-
lows
immediately
from
Lemma
3.15,
(i),
that
α
∈
Out
F
(Π
n
)[T
:
{|C|}].
Finally,
if
condition
(iv-2)
is
satisfied,
then,
by
applying
[CbTpI],
The-
orem
A,
(ii),
one
reduces
immediately
to
the
case
where
“n”
is
taken
to
be
n
−
1,
and
condition
(iv-1)
is
satisfied.
This
completes
the
proof
of
assertion
(iv).
Finally,
we
verify
assertion
(v).
First,
we
claim
that
the
following
assertion
holds:
Claim
3.16.B:
Out
F
(Π
n
)
=
Out
F
(Π
n
)[T
].
Indeed,
to
verify
Claim
3.16.B,
by
reordering
the
factors
of
X
n
,
we
may
assume
without
loss
of
generality
that
E
=
{1,
2,
3}.
Let
α
∈
F
Aut
(Π
n
).
Then
since
n
≥
3,
it
follows
immediately
from
[CbTpI],
Theorem
A,
(ii),
together
with
Lemma
3.2,
(iv),
that
the
outomorphism
preserves
the
Π
2/1
-conjugacy
class
of
cuspidal
of
Π
2/1
determined
by
α
subgroups
of
Π
2/1
associated
to
the
[unique
—
cf.
Lemma
3.2,
(ii)]
diagonal
cusp.
Thus,
it
follows
immediately
from
assertion
(iv)
in
the
case
where
condition
(iv-3)
is
satisfied
that
the
outomorphism
of
Π
3
determined
by
α
preserves
the
Π
3
-conjugacy
class
of
T
⊆
Π
3
.
This
completes
the
proof
of
Claim
3.16.B.
Next,
we
claim
that
the
following
assertion
holds:
Claim
3.16.C:
Out
F
(Π
n
)[T
]
=
Out
F
(Π
n
)[T
:
{Δ}].
Indeed,
since
n
≥
3,
this
follows
immediately
from
Theorem
2.3,
(iv),
together
with
a
similar
argument
to
the
argument
used
in
the
proof
of
[CmbCsp],
Corollary
3.4,
(i).
This
completes
the
proof
of
Claim
3.16.C.
Now
it
follows
immediately
from
Claims
3.16.B,
3.16.C
that
we
have
an
equality
Out
F
(Π
n
)
=
Out
F
(Π
n
)[T
:
{Δ}].
Thus,
it
follows
from
assertion
(ii)
and
the
first
displayed
equality
of
assertion
(iii),
together
COMBINATORIAL
ANABELIAN
TOPICS
II
85
with
Theorem
2.3,
(ii),
that,
to
complete
the
proof
of
the
content
of
the
first
two
displays
of
assertion
(v),
it
suffices
to
verify
the
equality
Out
FC
(Π
n
)
=
Out
FC
(Π
n
)[T
:
{C}].
On
the
other
hand,
this
follows
im-
mediately
from
the
portion
of
Lemma
3.15,
(i),
concerning
α|
T
.
[Note
that
one
verifies
easily
that
every
central
tripod
arises
from
a
cusp.]
Thus,
it
remains
to
verify
the
equality
of
the
final
display
of
assertion
(v).
In
light
of
what
has
already
been
verified
[cf.
also
Theorem
2.3,
(ii)],
to
verify
the
final
equality
of
assertion
(v),
it
suffices
to
verify
the
condition
“+”
on
the
right-hand
side
of
this
equality.
On
the
other
hand,
it
follows
immediately
—
by
replacing
an
element
of
the
left-hand
side
of
the
equality
under
consideration
by
a
composite
of
the
element
with
a
suitable
outomorphism
arising
from
an
element
of
Out
FC
(Π
4
)
[cf.
the
equality
of
the
first
display
of
assertion
(v)]
—
from
[CmbCsp],
Lemma
2.4,
that
it
suffices
to
verify
the
condition
“+”
on
an
element
of
the
left-hand
side
of
the
equality
under
consideration
that
induces
the
identity
automorphism
on
Cusp(G).
Then
the
equality
under
consideration
follows
immediately,
in
light
of
the
assumption
that
r
=
0,
by
first
applying
Lemma
3.15,
(i)
[in
the
case
where
we
take
the
“E”
of
loc.
cit.
to
be
a
subset
of
E
of
cardinality
two,
and
we
apply
the
argument
involving
specialization
isomorphisms
applied
in
the
proof
of
assertion
(iv)],
and
then
applying
Lemma
3.15,
(i),
(ii)
[in
the
case
where
we
take
the
“E”
of
loc.
cit.
to
be
E].
This
completes
the
proof
of
assertion
(v).
Remark
3.16.1.
Theorem
3.16,
(i),
may
be
regarded
as
a
general-
ization
of
[CmbCsp],
Corollary
1.10,
(ii).
On
the
other
hand,
Theo-
rem
3.16,
(v),
may
be
regarded
as
a
more
precise
version
of
[CmbCsp],
Corollary
3.4.
Theorem
3.17
(Synchronization
of
tripods
in
two
dimensions).
In
the
notation
of
Theorem
3.16,
suppose
that
n
=
2,
and
that
#E
=
1;
thus,
one
may
regard
the
E-tripod
T
of
Π
n
as
a
verticial
subgroup
∼
of
Π
E
→
Π
G
associated
to
a
vertex
v
T
∈
Vert(G)
of
type
(0,
3)
[cf.
Definition
3.1,
(ii)].
Let
E
⊆
{1,
·
·
·
,
n}
and
T
⊆
Π
E
an
E
-tripod
of
Π
n
.
Then
the
following
hold:
(i)
Suppose
that
there
exists
an
edge
e
∈
E(v
T
)
from
which
T
arises
[cf.
Definition
3.7,
(i)].
[Thus,
it
holds
that
E
=
{1,
2}.]
Then
it
holds
that
Out
FC
(Π
n
)[T
:
{|C|,
Δ}]
⊆
Out
FC
(Π
n
)[T
:
{|C|,
Δ,
+}]
[cf.
the
notational
conventions
of
Theorem
3.16,
(i)].
More-
over,
there
exists
a
geometric
[cf.
Definition
3.4,
(ii)]
outer
86
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
isomorphism
T
→
T
such
that
the
diagram
Out
FC
(Π
n
)[T
:
{|C|,
Δ}]
Out
FC
(Π
n
)[T
:
{|C|,
Δ,
+}]
⊆
T
T
↓
↓
T
T
∼
−→
Out(T
)
Out(T
)
[cf.
the
notation
of
Theorem
3.16,
(i)]
—
where
the
lower
horizontal
arrow
is
the
isomorphism
induced
by
the
outer
iso-
∼
morphism
in
question
T
→
T
—
commutes.
(ii)
Suppose
that
#E
=
1.
Thus,
one
may
regard
the
E
-tripod
∼
T
of
Π
n
as
a
verticial
subgroup
of
Π
E
→
Π
G
associated
to
a
vertex
v
T
∈
Vert(G)
of
type
(0,
3).
Suppose,
moreover,
that
N
(v
T
)
∩
N
(v
T
)
=
∅.
Then
there
exists
a
geometric
[cf.
Def-
∼
inition
3.4,
(ii)]
outer
isomorphism
T
→
T
such
that
if
we
write
Out
FC
(Π
n
)[T,
T
:
{|C|,
Δ}]
=
Out
FC
(Π
n
)[T
:
{|C|,
Δ}]
∩
Out
FC
(Π
n
)[T
:
{|C|,
Δ}]
,
def
then
the
diagram
Out
FC
(Π
n
)[T,
T
:
{|C|,
Δ}]
⏐
⏐
T
T
Out(T
)
Out
FC
(Π
n
)[T,
T
:
{|C|,
Δ}]
⏐
⏐
T
T
∼
−−−→
Out(T
)
—
where
the
lower
horizontal
arrow
is
the
isomorphism
induced
∼
by
the
outer
isomorphism
in
question
T
→
T
—
commutes.
Proof.
First,
we
verify
assertion
(i).
Let
us
observe
that
the
inclu-
sion
Out
FC
(Π
n
)[T
:
{|C|}]
⊆
Out
FC
(Π
n
)[T
],
hence
also
the
inclusion
Out
FC
(Π
n
)[T
:
{|C|,
Δ}]
⊆
Out
FC
(Π
n
)[T
],
follows
immediately
from
Theorem
3.16,
(iv),
in
the
case
where
condition
(iv-1)
is
satisfied.
Thus,
one
verifies
easily
from
Lemma
3.15,
(i),
(ii)
[cf.
also
Lemma
3.14,
(iv)],
that
the
remainder
of
assertion
(i)
holds.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
It
follows
immediately
from
[CmbCsp],
Proposition
1.2,
(iii),
that
we
may
assume
without
loss
of
generality
that
E
=
E.
Write
T
⊆
Π
n
for
the
{1,
2}-tripod
of
Π
n
arising
from
e
∈
N
(v
T
)
∩
N
(v
T
).
Then
it
follows
from
assertion
∼
∼
(i)
that
there
exist
geometric
outer
isomorphisms
T
→
T
,
T
→
T
that
satisfy
the
condition
of
assertion
(i)
[i.e.,
for
the
pairs
(T,
T
)
and
(T
,
T
)].
Thus,
one
verifies
easily
that
the
[necessarily
geometric]
outer
∼
∼
isomorphism
T
→
T
←
T
obtained
by
forming
the
composite
of
these
two
outer
isomorphisms
satisfies
the
condition
of
assertion
(ii).
This
completes
the
proof
of
assertion
(ii).
COMBINATORIAL
ANABELIAN
TOPICS
II
87
Theorem
3.18
(Synchronization
of
tripods
in
three
or
more
dimensions).
In
the
notation
of
Theorem
3.16,
suppose
that
n
≥
3.
Then
the
following
hold:
(i)
It
holds
that
Out
FC
(Π
n
)[T
:
{|C|}]
=
Out
FC
(Π
n
)[T
:
{|C|,
Δ}]
[cf.
the
notational
conventions
of
Theorem
3.16,
(i)].
If,
more-
over,
n
≥
4
or
r
=
0,
then
it
holds
that
Out
FC
(Π
n
)[T
:
{|C|}]
=
Out
FC
(Π
n
)[T
:
{|C|,
Δ,
+}]
[cf.
the
notational
conventions
of
Theorem
3.16,
(i)].
(ii)
Let
E
⊆
{1,
·
·
·
,
n}
and
T
⊆
Π
E
an
E
-tripod
of
Π
n
.
Then
there
exists
a
geometric
[cf.
Definition
3.4,
(ii)]
outer
iso-
∼
morphism
T
→
T
such
that
if
we
write
Out
FC
(Π
n
)[T,
T
:
{|C|}]
=
Out
FC
(Π
n
)[T
:
{|C|}]
∩
Out
FC
(Π
n
)[T
:
{|C|}]
,
then
the
diagram
def
Out
FC
(Π
n
)[T,
T
:
{|C|}]
⏐
⏐
T
T
Out(T
)
Out
FC
(Π
n
)[T,
T
:
{|C|}]
⏐
⏐
T
T
∼
−−−→
Out(T
)
[cf.
the
notation
of
Theorem
3.16,
(i)]
—
where
the
lower
horizontal
arrow
is
the
isomorphism
induced
by
the
outer
iso-
∼
morphism
in
question
T
→
T
—
commutes.
Proof.
First,
we
verify
the
first
displayed
equality
of
assertion
(i).
Ob-
serve
that
it
follows
immediately
from
Lemma
3.8,
(i),
together
with
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
first
dis-
played
equality
of
Theorem
3.16,
(iii),
that
we
may
assume
without
loss
of
generality
that
T
is
E-strict,
which
thus
implies
that
#E
∈
{1,
2,
3}
[cf.
Lemma
3.8,
(ii)].
Now
we
apply
induction
on
3
−
#E
∈
{0,
1,
2}.
If
3
−
#E
=
0,
i.e.,
T
is
central
[cf.
Lemma
3.8,
(ii)],
then
the
first
dis-
played
equality
of
assertion
(i)
follows
immediately
from
Theorem
3.16,
(v).
Now
suppose
that
3
−
#E
>
0,
and
that
the
induction
hypoth-
esis
is
in
force.
Let
α
∈
Out
FC
(Π
n
)[T
:
{|C|}].
Then
it
follows
im-
mediately
from
Lemma
3.15,
(i),
(ii)
[cf.
also
conditions
(1),
(2)
of
Lemma
3.15,
(ii),
where
we
note
that
the
E,
E
,
T
,
T
of
the
present
discussion
correspond,
respectively,
to
the
“E
\
{i}”,
“E”,
“Π
v
”,
“T
”
of
Lemma
3.15],
that
there
exist
a
subset
E
⊆
E
⊆
{1,
·
·
·
,
n}
and
an
E
-tripod
T
⊆
Π
E
such
that
3
−
#E
<
3
−
#E,
T
⊆
Π
E
is
E
-strict,
and
α
∈
Out
FC
(Π
n
)[T
:
{|C|}]
[cf.
Lemma
3.15,
(i)].
Thus,
it
follows
immediately
from
the
induction
hypothesis
that
α
∈
Out
FC
(Π
n
)[T
:
{|C|,
Δ}].
In
particular,
it
follows
immediately
from
Lemma
3.15,
(ii),
88
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
that
—
for
a
suitable
choice
of
the
pair
(E
,
T
)
[cf.
the
statement
of
Lemma
3.15,
(ii)]
—
the
actions
of
α
on
T
and
T
may
be
related
by
means
of
a
geometric
outer
isomorphism,
which
thus
implies
that
α
∈
Out
FC
(Π
n
)[T
:
{|C|,
Δ}]
[cf.
Remark
3.4.1].
This
completes
the
proof
of
the
first
displayed
equality
of
assertion
(i).
Next,
we
verify
assertion
(ii).
First,
we
claim
that
the
following
assertion
holds:
Claim
3.18.A:
If
both
T
and
T
are
central,
then
the
pair
(T,
T
)
satisfies
the
property
stated
in
assertion
(ii).
Indeed,
this
assertion
follows
immediately
from
the
commutativity
of
the
displayed
diagram
of
Theorem
3.16,
(i).
Next,
we
claim
that
the
following
assertion
holds:
Claim
3.18.B:
Suppose
that
T
is
E-strict,
and
that
#E
=
3
[i.e.,
#E
∈
{1,
2}
—
cf.
Lemma
3.8,
(ii)].
Then
there
exist
a
subset
E
E
⊆
{1,
·
·
·
,
n}
and
an
E
-tripod
T
⊆
Π
E
such
that
T
is
E
-strict,
Out
FC
(Π
n
)[T
:
{|C|}]
⊆
Out
FC
(Π
n
)[T
:
{|C|}],
and,
moreover,
the
pair
(T,
T
)
satisfies
the
property
stated
in
assertion
(ii)
[i.e.,
where
one
takes
“T
”
to
be
T
].
Indeed,
this
follows
immediately
from
Lemma
3.15,
(i),
(ii)
[cf.
also
conditions
(1),
(2)
of
Lemma
3.15,
(ii),
where
we
note
that
the
E,
E
,
T
,
T
of
the
present
discussion
correspond,
respectively,
to
the
“E
\
{i}”,
“E”,
“Π
v
”,
“T
”
of
Lemma
3.15],
together
with
the
first
displayed
equality
of
assertion
(i).
This
completes
the
proof
of
Claim
3.18.B.
To
verify
assertion
(ii),
let
us
observe
that
it
follows
immediately
from
Lemma
3.8,
(i),
together
with
a
similar
argument
to
the
argument
applied
in
the
proof
of
the
first
displayed
equality
of
Theorem
3.16,
(iii),
that
we
may
assume
without
loss
of
generality
that
T
is
E-strict;
in
par-
ticular,
#E
∈
{1,
2,
3}
[cf.
Lemma
3.8,
(ii)].
Next,
let
us
observe
that,
by
comparing
two
arbitrary
tripods
of
Π
n
to
a
fixed
central
tripod
of
Π
n
[and
applying
Theorem
3.16,
(v)],
one
may
reduce
immediately
to
the
case
where
T
is
central.
Moreover,
by
successive
application
of
Claim
3.18.B,
one
reduces
immediately
to
the
case
where
both
T
and
T
are
central,
which
was
verified
in
Claim
3.18.A.
This
completes
the
proof
of
assertion
(ii).
Finally,
the
second
displayed
equality
of
assertion
(i)
follows
immediately
from
assertion
(ii),
together
with
Theorem
3.16,
(v).
This
completes
the
proof
of
Theorem
3.18.
Definition
3.19.
Suppose
that
n
≥
3.
Let
us
write
Π
tpd
COMBINATORIAL
ANABELIAN
TOPICS
II
89
for
the
i-central
E-tripod
of
Π
n
[cf.
Definitions
3.3,
(i);
3.7,
(ii)],
where
E
⊆
{1,
.
.
.
,
n}
is
a
subset
of
cardinality
3,
and
i
∈
E.
Then
it
follows
from
Theorem
3.16,
(i),
(v),
that
one
has
a
natural
homomorphism
T
Π
tpd
:
Out
FC
(Π
n
)
=
Out
FC
(Π
n
)[Π
tpd
:
{|C|,
Δ}]
−→
Out
C
(Π
tpd
)
Δ
[cf.
Definition
3.4,
(i)],
which
is
in
fact
independent
of
E
and
i
[cf.
Theorem
3.16,
(i)].
We
shall
refer
to
this
homomorphism
as
the
tripod
homomorphism
associated
to
Π
n
and
write
Out
FC
(Π
n
)
geo
⊆
Out
FC
(Π
n
)
for
the
kernel
of
this
homomorphism
[cf.
Remark
3.19.1
below].
Note
that
it
follows
from
Theorem
3.16,
(v),
that
if
n
≥
4
or
r
=
0,
then
the
image
of
the
tripod
homomorphism
is
contained
in
Out
C
(Π
tpd
)
Δ+
⊆
Out
C
(Π
tpd
)
Δ
[cf.
Definition
3.4,
(i)].
If
n
≥
4
or
r
=
0,
then
T
Π
tpd
may
also
be
regarded
as
a
homomorphism
defined
on
Out
F
(Π
n
)
(=
Out
FC
(Π
n
)
—
cf.
Theorem
2.3,
(ii));
in
this
case,
we
shall
write
def
Out
F
(Π
n
)
geo
=
Out
FC
(Π
n
)
geo
.
Remark
3.19.1.
Let
us
recall
that
if
we
write
π
1
((M
g,[r]
)
Q
)
for
the
étale
fundamental
group
of
the
moduli
stack
(M
g,[r]
)
Q
of
hyperbolic
curves
of
type
(g,
r)
over
Q
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”],
then
we
have
a
natural
outer
homo-
morphism
π
1
((M
g,[r]
)
Q
)
−→
Out
FC
(Π
n
)
.
Suppose
that
n
≥
4.
Then
Out
FC
(Π
n
)
=
Out
F
(Π
n
)
does
not
de-
pend
on
n
[cf.
Theorem
2.3,
(ii);
[NodNon],
Theorem
B].
Morever,
one
verifies
easily
that
the
image
of
the
geometric
fundamental
group
π
1
((M
g,[r]
)
Q
)
⊆
π
1
((M
g,[r]
)
Q
)
—
where
we
use
the
notation
Q
to
denote
an
algebraic
closure
of
Q
—
via
the
above
displayed
outer
homomor-
phism
is
contained
in
the
kernel
Out
FC
(Π
n
)
geo
⊆
Out
FC
(Π
n
)
of
the
tripod
homomorphism
associated
to
Π
n
[cf.
Definition
3.19].
Thus,
the
outer
homomorphism
of
the
above
display
fits
into
a
commutative
diagram
of
profinite
groups
1
−−−→
π
1
((M
g,[r]
)
Q
)
−−−→
π
1
((M
g,[r]
)
Q
)
−−−→
⏐
⏐
⏐
⏐
1
−−−→
Out
F
(Π
n
)
geo
−−−→
Out
F
(Π
n
)
Gal(Q/Q)
⏐
⏐
−−−→
1
T
tpd
Π
−−
−→
Out
C
(Π
tpd
)
Δ+
—
where
the
horizontal
sequences
are
exact.
In
§4
below,
we
shall
ver-
ify
that
the
lower
right-hand
horizontal
arrow
T
Π
tpd
is
surjective
[cf.
Corollary
4.15
below].
On
the
other
hand,
if
Σ
is
the
set
of
all
prime
numbers,
then
it
follows
from
Belyi’s
Theorem
that
the
right-hand
vertical
arrow
is
injective;
moreover,
the
surjectivity
of
the
right-hand
90
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
vertical
arrow
has
been
conjectured
in
the
theory
of
the
Grothendieck-
Teichmüller
group.
From
this
point
of
view,
one
may
regard
the
quo-
T
Πtpd
tient
Out
F
(Π
n
)
Out
C
(Π
tpd
)
Δ+
as
a
sort
of
arithmetic
quotient
of
Out
F
(Π
n
)
and
the
subgroup
Out
F
(Π
n
)
geo
⊆
Out
F
(Π
n
)
as
a
sort
of
geometric
portion
of
Out
F
(Π
n
).
Definition
3.20.
Let
m
be
a
positive
integer
and
Y
log
a
stable
log
curve
over
(Spec
k)
log
.
For
each
nonnegative
integer
i,
write
Y
Π
i
for
the
“Π
i
”
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
Y
log
.
Then
we
shall
say
that
an
isomorphism
(respectively,
outer
isomorphism)
∼
Π
1
→
Y
Π
1
is
m-cuspidalizable
if
it
arises
from
a
[necessarily
unique,
up
to
a
permutation
of
the
m
factors,
by
[NodNon],
Theorem
B]
PFC-
∼
admissible
[cf.
[CbTpI],
Definition
1.4,
(iii)]
isomorphism
Π
m
→
Y
Π
m
.
Proposition
3.21
(Tripod
homomorphisms
and
finite
étale
cov-
erings).
Let
Y
log
be
a
stable
log
curve
over
(Spec
k)
log
and
Y
log
→
X
log
a
finite
log
étale
covering
over
(Spec
k)
log
.
For
each
positive
inte-
ger
i,
write
Y
i
log
(respectively,
Y
Π
i
)
for
the
“X
i
log
”
(respectively,
“Π
i
”)
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
Y
log
.
Suppose
that
Y
log
→
X
log
is
geometrically
pro-Σ
and
geometrically
Galois,
i.e.,
Y
log
→
X
log
determines
an
injection
Y
Π
1
→
Π
1
[that
is
well-
defined
up
to
Π
1
-conjugation]
whose
image
is
normal.
Let
α
be
an
Y
automorphism
of
Π
1
that
preserves
Π
1
⊆
Π
1
.
Suppose,
moreover,
that
is
n-cuspidalizable
[cf.
the
outomorphism
α
of
Π
1
determined
by
α
Definition
3.20].
Then
the
following
hold:
(i)
The
outomorphism
Y
α
of
Y
Π
1
determined
by
α
is
n-cuspidali-
zable
[cf.
Definition
3.20].
(ii)
Suppose
that
n
≥
3.
Let
Π
tpd
⊆
Π
3
,
Y
Π
tpd
⊆
Y
Π
3
be
1-central
[{1,
2,
3}-]tripods
[cf.
Definitions
3.3,
(i);
3.7,
(ii)]
of
Π
n
,
Y
Π
n
,
respectively.
Write
α
n
,
Y
α
n
for
the
respective
FC-admissible
outomorphisms
of
Π
n
,
Y
Π
n
determined
by
the
n-cuspidalizable
outomorphisms
α,
Y
α
[cf.
(i)].
Then
there
exists
a
geometric
∼
[cf.
Definition
3.4,
(ii)]
outer
isomorphism
φ
tpd
:
Π
tpd
→
Y
Π
tpd
such
that
the
outomorphism
T
Π
tpd
(α
n
)
[cf.
Definition
3.19]
of
Π
tpd
is
compatible
with
the
outomorphism
T
Y
Π
tpd
(
Y
α
n
)
[cf.
Definition
3.19]
of
Y
Π
tpd
relative
to
φ
tpd
.
Proof.
First,
let
us
observe
that,
to
verify
Proposition
3.21
—
by
apply-
ing
a
suitable
specialization
isomorphism
[cf.
the
discussion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1]
—
we
may
assume
without
loss
of
generality
that
X
log
and
Y
log
are
smooth
log
curves
over
(Spec
k)
log
.
Write
(U
X
)
n
,
(U
Y
)
n
for
the
[open
subschemes
COMBINATORIAL
ANABELIAN
TOPICS
II
91
of
X
n
,
Y
n
determined
by
the]
1-interiors
[cf.
[MzTa],
Definition
5.1,
(i)]
of
X
n
log
,
Y
n
log
,
respectively.
[Here,
we
note
that
in
the
present
situ-
ation,
the
0-interior
of
(Spec
k)
log
,
hence
also
of
X
n
log
,
Y
n
log
,
is
empty!]
def
def
Thus,
one
verifies
easily
that
U
X
=
(U
X
)
1
,
U
Y
=
(U
Y
)
1
are
hyper-
bolic
curves
over
k,
and
that
(U
X
)
n
,
(U
Y
)
n
are
naturally
isomorphic
to
the
n-th
configuration
spaces
of
U
X
,
U
Y
,
respectively.
Write
U
X
×n
,
U
Y
×n
for
the
respective
fiber
products
of
n
copies
of
U
X
,
U
Y
over
k;
Π
×n
1
,
Y
×n
Y
Π
1
for
the
respective
direct
products
of
n
copies
of
Π
1
,
Π
1
;
V
n
for
the
fiber
product
of
the
natural
open
immersion
(U
X
)
n
→
U
X
×n
and
the
natural
finite
étale
covering
U
Y
×n
→
U
X
×n
.
Then
one
verifies
easily
that
the
resulting
open
immersion
V
n
→
U
Y
×n
factors
through
the
nat-
ural
open
immersion
(U
Y
)
n
→
U
Y
×n
,
i.e.,
we
obtain
an
open
immersion
V
n
→
(U
Y
)
n
.
That
is
to
say,
whereas
(U
Y
)
n
is
the
open
subscheme
of
U
Y
×n
obtained
by
removing
the
various
diagonals
of
U
Y
×n
,
the
scheme
V
n
may
be
thought
of
as
the
open
subscheme
of
U
Y
×n
obtained
by
removing
the
various
Galois
conjugates
of
these
diagonals,
relative
to
the
action
of
the
Galois
group
Gal(U
Y
×n
/U
X
×n
)
=
Gal(U
Y
/U
X
)
×n
.
In
particular,
we
obtain
a
natural
outer
isomorphism
and
outer
surjection
∼
Y
×n
Π
1
←
Π
V
n
Y
Π
n
Π
n
×
Π
×n
1
—
where
we
write
Π
V
n
for
the
maximal
pro-Σ
quotient
of
the
étale
fundamental
group
of
V
n
.
Now
we
verify
assertion
(i).
Let
α
n
be
an
FC-admissible
automor-
of
Π
1
with
respect
phism
of
Π
n
that
lies
over
the
automorphism
α
n
is
to
each
of
the
n
natural
projections
Π
n
Π
1
.
Then
since
α
FC-admissible
and
commutes
with
the
image
of
the
natural
inclusion
S
n
→
Out(Π
n
)
[cf.
[NodNon],
Theorem
B],
one
verifies
easily,
in
light
of
the
description
given
above
of
V
n
,
that
the
outomorphism
of
Y
×n
Π
n
×
Π
×n
Π
1
induced
by
α
n
and
Y
α
preserves
the
inertia
subgroups
1
associated
to
each
irreducible
component
of
the
complement
U
Y
×n
\
V
n
.
Thus,
since
[by
the
Zariski-Nagata
purity
theorem]
the
inertia
sub-
groups
of
the
irreducible
components
of
the
complement
(U
Y
)
n
\
V
n
normally
topologically
generate
the
kernel
of
the
above
outer
surjec-
tion
Π
V
n
Y
Π
n
,
we
conclude,
by
applying
the
morphisms
of
the
above
Y
×n
display,
that
the
outomorphism
of
Π
n
×
Π
×n
Π
1
induced
by
α
n
and
1
Y
Y
α
determines
an
FC-admissible
outomorphism
of
Π
n
.
Moreover,
one
verifies
easily
that
the
resulting
outomorphism
of
Y
Π
n
lies
over
the
outomorphism
Y
α
of
Y
Π
1
.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
First,
let
us
observe
that
the
natural
inclusion
Π
tpd
→
Π
3
,
together
with
the
trivial
homomorphism
Π
tpd
→
({1}
→)
Y
Π
×3
1
[cf.
Definition
3.3,
(ii);
Lemma
3.6,
(v);
Definition
3.7,
Y
×3
∼
(ii)],
determines
an
injection
Π
tpd
→
Π
3
×
Π
×3
Π
1
←
Π
V
3
.
Moreover,
1
it
follows
immediately
from
the
fact
that
the
blow-up
operation
that
gives
rise
to
a
central
tripod
is
compatible
with
étale
localization
[cf.
the
92
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
discussion
of
[CmbCsp],
Definition
1.8]
that
—
after
possibly
replacing
Y
tpd
Π
⊆
Y
Π
3
by
a
suitable
Y
Π
3
-conjugate
of
Y
Π
tpd
—
the
composite
of
this
injection
Π
tpd
→
Π
V
3
with
the
natural
outer
surjection
Π
V
3
Y
Π
3
of
the
above
display
determines
a
geometric
outer
[cf.
Lemma
3.12,
∼
(i)]
isomorphism
φ
tpd
:
Π
tpd
→
Y
Π
tpd
⊆
Y
Π
3
.
On
the
other
hand,
one
verifies
easily
[cf.
the
construction
of
Y
α
n
given
in
the
proof
of
assertion
(i)]
that
this
outer
isomorphism
φ
tpd
satisfies
the
property
stated
in
assertion
(ii).
This
completes
the
proof
of
assertion
(ii).
Corollary
3.22
(Non-surjectivity
result).
In
the
notation
of
The-
orem
3.16,
suppose
that
(g,
r)
∈
{(0,
3);
(1,
1)}.
Then
the
natural
in-
jection
Out
FC
(Π
2
)
→
Out
FC
(Π
1
)
of
[NodNon],
Theorem
B,
is
not
surjective.
Proof.
First,
let
us
observe
—
by
considering
a
suitable
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
and
applying
a
suitable
special-
ization
isomorphism
[cf.
the
discussion
preceding
[CmbCsp],
Defini-
tion
2.1,
as
well
as
[CbTpI],
Remark
5.6.1]
—
that,
to
verify
Corol-
lary
3.22,
we
may
assume
without
loss
of
generality
that
G
is
totally
degenerate
[cf.
[CbTpI],
Definition
2.3,
(iv)],
i.e.,
that
every
vertex
of
G
is
a
tripod
of
X
n
log
[cf.
Definition
3.1,
(v)].
Note
that
[since
(g,
r)
∈
{(0,
3);
(1,
1)}]
this
implies
that
#Vert(G)
≥
2.
Let
us
fix
def
a
vertex
v
0
∈
Vert(G)
and
write
α
v
0
=
id
G|
v
0
∈
Aut
|grph|
(G|
v
0
)
[cf.
[CbTpI],
Definitions
2.1,
(iii),
and
2.6,
(i);
Remark
4.1.2
of
the
present
monograph].
For
each
v
∈
Vert(G)
\
{v
0
},
let
α
v
∈
Aut
|grph|
(G|
v
)
be
a
nontrivial
automorphism
of
G|
v
such
that
α
v
∈
Out
C
(Π
G|
v
)
Δ
,
and,
moreover,
χ
G|
v
(α
v
)
=
1
[cf.
[CbTpI],
Definition
3.8,
(ii)].
Here,
we
note
that
since
the
image
of
the
natural
outer
Galois
representation
of
the
absolute
Galois
group
of
Q
associated
to
P
1
Q
\
{0,
1,
∞}
is
contained
in
“Out
C
(−)
Δ
”,
by
considering
a
nontrivial
element
of
this
image
whose
image
via
the
cyclotomic
character
is
trivial,
one
verifies
immediately
[e.g.,
by
applying
[LocAn],
Theorem
A]
that
such
an
automorphism
α
v
∈
Aut
|grph|
(G|
v
)
always
exists.
Then
it
follows
immediately
from
[CbTpI],
Theorem
B,
(iii),
that
there
exists
an
automorphism
α
∈
Aut
|grph|
(G)
such
that
ρ
Vert
G
(α)
=
(α
v
)
v∈Vert(G)
.
Now
assume
that
there
exists
an
outomorphism
α
2
∈
Out
FC
(Π
2
)
such
that
α
∈
Aut
|grph|
(G)
∼
(⊆
Out(Π
G
)
←
Out(Π
1
))
is
equal
to
the
image
of
α
2
via
the
injection
in
question
Out
FC
(Π
2
)
→
Out
FC
(Π
1
).
Then,
for
each
v
∈
Vert(G),
since
α
v
∈
Out
C
(Π
G|
v
)
Δ
,
and
α
∈
Aut
|grph|
(G),
it
follows
immediately
from
the
various
definitions
involved
that
α
2
∈
Out
FC
(Π
2
)[Π
v
:
{|C|,
Δ}]
—
where
we
use
the
notation
Π
v
to
denote
a
verticial
subgroup
of
COMBINATORIAL
ANABELIAN
TOPICS
II
∼
93
def
Π
G
←
Π
1
associated
to
v
∈
Vert(G).
Thus,
since
α
v
0
=
id
G|
v
0
,
it
fol-
lows
from
Theorem
3.17,
(ii),
that
α
v
=
id
G|
v
for
every
v
∈
Vert(G),
in
contradiction
to
the
fact
that
for
v
∈
Vert(G)
\
{v
0
}
(
=
∅),
the
auto-
morphism
α
v
∈
Aut
|grph|
(G|
v
)
is
nontrivial.
This
completes
the
proof
of
Corollary
3.22.
Remark
3.22.1.
(i)
Let
us
recall
from
[NodNon],
Corollary
6.6,
that,
in
the
dis-
crete
case,
the
homomorphism
that
corresponds
to
the
homo-
morphism
discussed
in
Corollary
3.22
is,
in
fact,
surjective;
moreover,
this
surjectivity
may
be
regarded
as
an
immediate
consequence
of
the
Dehn-Nielsen-Baer
theorem
—
cf.
the
proof
of
[CmbCsp],
Theorem
5.1,
(ii).
This
phenomenon
illustrates
that,
in
general,
analogous
constructions
in
the
discrete
and
profinite
cases
may
in
fact
exhibit
quite
different
behavior.
(ii)
In
the
context
of
(i),
we
recall
another
famous
example
of
sub-
stantially
different
behavior
in
the
discrete
and
profinite
cases:
As
is
well-known,
in
classical
algebraic
topology,
singular
co-
homology
with
coefficients
in
Z
yields
a
“good”
cohomology
theory
with
coefficients
in
Z.
On
the
other
hand,
in
the
1960’s,
Serre
gave
an
argument
involving
supersingular
elliptic
curves
in
characteristic
p
>
0
which
shows
that
such
a
“good”
coho-
mology
theory
with
coefficients
in
Z
[or
even
in
Z
p
!]
cannot
exist
for
smooth
varieties
of
positive
characteristic.
(iii)
In
[Lch],
various
conjectures
concerning
[in
the
notation
of
the
present
monograph]
the
profinite
group
“Out(Π
1
)”
were
intro-
duced.
However,
at
the
time
of
writing,
the
authors
of
the
present
monograph
were
unable
to
find
any
justification
for
the
validity
of
these
conjectures
that
goes
beyond
the
observa-
tion
that
the
discrete
analogues
of
these
conjectures
are
indeed
valid.
That
is
to
say,
there
does
not
appear
to
exist
any
justi-
fication
for
excluding
the
possibility
that
—
just
as
in
the
case
of
the
examples
discussed
in
(i),
(ii),
i.e.,
the
Dehn-Nielsen-
Baer
theorem
and
singular
cohomology
with
coefficients
in
Z
—
the
discrete
and
profinite
cases
exhibit
substantially
differ-
ent
behavior.
In
particular,
it
appears
to
the
authors
that
it
is
desirable
that
this
issue
be
addressed
in
a
satisfactory
fashion
in
the
context
of
these
conjectures.
Remark
3.22.2.
As
discussed
in
Remark
3.22.1,
(i),
in
the
discrete
case,
the
homomorphism
that
corresponds
to
the
homomorphism
dis-
cussed
in
Corollary
3.22
is,
in
fact,
bijective.
The
proof
of
Corollary
3.22
94
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
fails
in
the
discrete
case
for
the
following
reason:
The
pro-Σ
“Π
1
”
of
a
tripod
admits
nontrivial
C-admissible
outomorphisms
that
commute
with
the
outer
modular
symmetries
and,
moreover,
lie
in
the
kernel
of
the
cyclotomic
character
[cf.
the
proof
of
Corollary
3.22].
By
contrast,
the
discrete
“Π
1
”
of
a
tripod
does
not
admit
such
outomorphisms.
In-
deed,
it
follows
from
a
classical
result
of
Nielsen
[cf.
[CmbCsp],
Remark
5.3.1]
that
the
discrete
“Out
C
(Π
1
)
cusp
”
in
the
case
of
a
tripod
is
a
finite
group
of
order
2
whose
unique
nontrivial
element
arises
from
complex
conjugation.
Remark
3.22.3.
It
follows
from
[NodNon],
Theorem
B,
together
with
Corollary
3.22,
that
if
(g,
r)
∈
{(0,
3);
(1,
1)},
then
the
homomorphism
Out
FC
(Π
n+1
)
→
Out
FC
(Π
n
)
of
[NodNon],
Theorem
B,
fits
into
the
following
sequences
of
homomorphisms
of
profinite
groups:
If
r
=
0,
then
for
any
n
≥
3,
≃
≃
?
∼
Out
FC
(Π
n
)
→
Out
FC
(Π
3
)
→
Out
FC
(Π
2
)
→
Out
FC
(Π
1
)
.
If
r
=
0,
then
for
any
n
≥
4,
∼
≃
?
≃
≃
?
Out
FC
(Π
n
)
→
Out
FC
(Π
4
)
→
Out
FC
(Π
3
)
→
Out
FC
(Π
2
)
→
Out
FC
(Π
1
)
.
Definition
3.23.
Let
Σ
0
be
a
nonempty
set
of
prime
numbers
and
G
0
a
semi-graph
of
anabelioids
of
pro-Σ
0
PSC-type.
Write
Π
G
0
for
the
[pro-Σ
0
]
fundamental
group
of
G
0
.
(i)
Let
H
be
a
semi-graph
of
anabelioids
of
pro-Σ
0
PSC-type,
∼
S
⊆
Node(H),
and
φ
:
H
S
→
G
0
[cf.
[CbTpI],
Definition
2.8,
for
more
on
this
notation]
an
isomorphism
[of
semi-graphs
of
anabelioids
of
PSC-type].
Then
we
shall
refer
to
the
triple
(H,
S,
φ)
as
a
degeneration
structure
on
G
0
.
(ii)
Let
(H
1
,
S
1
,
φ
1
),
(H
2
,
S
2
,
φ
2
)
be
two
degeneration
structures
on
G
0
[cf.
(i)].
Then
we
shall
write
(H
2
,
S
2
,
φ
2
)
(H
1
,
S
1
,
φ
1
)
if
there
exist
a
subset
S
2,1
⊆
S
2
of
S
2
and
a(n)
[uniquely
de-
termined,
by
φ
1
and
φ
2
!
—
cf.
[CmbGC],
Proposition
1.5,
∼
(ii)]
isomorphism
φ
2,1
:
(H
2
)
S
2,1
→
H
1
[i.e.,
a
degeneration
structure
(H
2
,
S
2,1
,
φ
2,1
)
on
H
1
]
such
that
φ
2,1
maps
S
2
\
S
2,1
bijectively
onto
S
1
,
and
the
diagram
∼
((H
2
)
S
2,1
)
S
2
\S
2,1
−−−→
(H
1
)
S
1
⏐
⏐
⏐
⏐
φ
1
(H
2
)
S
2
φ
2
−−−→
∼
G
0
COMBINATORIAL
ANABELIAN
TOPICS
II
95
—
where
the
upper
horizontal
arrow
is
the
isomorphism
in-
duced
by
φ
2,1
,
and
the
left-hand
vertical
arrow
is
the
natural
isomorphism
—
commutes.
[Here,
we
note
that
the
subset
S
2,1
is
also
uniquely
determined
by
φ
1
and
φ
2
—
cf.
[CmbGC],
Proposition
1.2,
(i).]
(iii)
Let
(H
1
,
S
1
,
φ
1
),
(H
2
,
S
2
,
φ
2
)
be
two
degeneration
structures
on
G
0
[cf.
(i)].
Then
we
shall
say
that
(H
1
,
S
1
,
φ
1
)
is
co-Dehn
to
(H
2
,
S
2
,
φ
2
)
if
there
exists
a
degeneration
structure
(H
3
,
S
3
,
φ
3
)
on
G
0
such
that
(H
3
,
S
3
,
φ
3
)
(H
1
,
S
1
,
φ
1
);
(H
3
,
S
3
,
φ
3
)
(H
2
,
S
2
,
φ
2
)
[cf.
(ii)].
(iv)
Let
(H,
S,
φ)
be
a
degeneration
structure
on
G
0
[cf.
(i)]
and
α
∈
Out(Π
G
0
).
Then
we
shall
say
that
α
is
an
(H,
S,
φ)-Dehn
multi-twist
of
G
0
if
α
is
contained
in
the
image
of
the
composite
∼
∼
Dehn(H)
→
Out(Π
H
)
←
Out(Π
H
S
)
→
Out(Π
G
0
)
—
where
the
first
arrow
is
the
natural
inclusion
[cf.
[CbTpI],
Definition
4.4],
the
second
arrow
is
the
isomorphism
deter-
mined
by
Φ
H
S
[cf.
[CbTpI],
Definition
2.10],
and
the
third
arrow
is
the
isomorphism
determined
by
φ.
We
shall
say
that
α
is
a
nondegenerate
(respectively,
positive
definite)
(H,
S,
φ)-
Dehn
multi-twist
of
G
0
if
α
is
the
image
of
a
nondegenerate
[cf.
[CbTpI],
Definition
5.8,
(ii)]
(respectively,
positive
definite
[cf.
[CbTpI],
Definition
5.8,
(iii)])
profinite
Dehn
multi-twist
of
H
via
the
above
composite.
(v)
Let
m
be
a
positive
integer
and
Y
log
a
stable
log
curve
over
(Spec
k)
log
.
If
m
≥
2,
then
suppose
that
Σ
0
is
either
equal
to
Primes
or
of
cardinality
one.
For
each
nonnegative
integer
i,
write
Y
Π
i
(respectively,
H)
for
the
“Π
i
”
(respectively,
“G”)
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
Y
log
.
Then
we
shall
say
that
a
degeneration
structure
(H,
S,
φ)
on
G
[cf.
(i)]
is
m-cuspidalizable
if
the
composite
Y
∼
Φ
H
S
∼
φ
∼
∼
Π
1
−→
Π
H
←−
Π
H
S
−→
Π
G
←−
Π
1
—
where
the
first
and
fourth
arrows
are
the
natural
outer
iso-
morphisms
[cf.
Definition
3.1,
(ii)],
and
the
second
arrow
Φ
H
S
is
the
natural
outer
isomorphism
of
[CbTpI],
Definition
2.10
—
is
m-cuspidalizable
[cf.
Definition
3.20].
Remark
3.23.1.
One
interesting
open
problem
in
the
theory
of
profi-
nite
Dehn
multi-twists
developed
in
[CbTpI],
§4,
is
the
following:
In
96
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
the
notation
of
Definition
3.23,
for
i
=
1,
2,
let
(H
i
,
S
i
,
φ
i
)
be
a
de-
generation
structure
on
G
0
[cf.
Definition
3.23,
(i)];
α
i
∈
Out(Π
G
0
)
a
nondegenerate
(H
i
,
S
i
,
φ
i
)-Dehn
multi-twist
[cf.
Definition
3.23,
(iv)].
Then:
Suppose
that
α
1
commutes
with
α
2
.
Then
is
(H
1
,
S
1
,
φ
1
)
co-Dehn
to
(H
2
,
S
2
,
φ
2
)
[cf.
Definition
3.23,
(iii)]?
It
is
not
clear
to
the
authors
at
the
time
of
writing
whether
or
not
this
question
may
be
answered
in
the
affirmative.
Nevertheless,
we
are
able
to
obtain
a
partial
result
in
this
direction
[cf.
Corollary
3.25
below].
Proposition
3.24
(Compatibility
of
tripod
homomorphisms).
Suppose
that
n
≥
3.
Then
the
following
hold:
(i)
Let
Y
log
be
a
stable
log
curve
over
(Spec
k)
log
.
For
each
non-
negative
integer
i,
write
Y
Π
i
(respectively,
H)
for
the
“Π
i
”
(re-
spectively,
“G”)
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
Y
log
.
Let
(H,
S,
φ)
be
an
n-cuspidalizable
degeneration
∼
structure
on
G
[cf.
Definition
3.23,
(i),
(v)];
φ
n
:
Y
Π
n
→
Π
n
a
PFC-admissible
outer
isomorphism
[cf.
[CbTpI],
Definition
1.4,
(iii)]
that
lies
over
the
displayed
composite
isomorphism
of
Definition
3.23,
(v);
Π
tpd
⊆
Π
3
,
Y
Π
tpd
⊆
Y
Π
3
1-central
[{1,
2,
3}-]tripods
[cf.
Definitions
3.3,
(i);
3.7,
(ii)]
of
Π
n
,
Y
Π
n
,
respectively.
Then
there
exists
an
outer
isomorphism
∼
φ
tpd
:
Y
Π
tpd
→
Π
tpd
such
that
the
diagram
∼
Out
FC
(
Y
Π
n
)
−−−→
Out
FC
(Π
n
)
⏐
⏐
⏐
T
⏐
T
Y
Πtpd
Πtpd
∼
Out(
Y
Π
tpd
)
−−−→
Out(Π
tpd
)
[cf.
Definition
3.19]
—
where
the
upper
and
lower
horizontal
arrows
are
the
isomorphisms
induced
by
φ
n
,
φ
tpd
,
respectively
—
commutes,
up
to
inner
automorphisms
of
Out(Π
tpd
).
In
particular,
φ
n
determines
an
isomorphism
∼
Out
FC
(
Y
Π
n
)
geo
−→
Out
FC
(Π
n
)
geo
[cf.
Definition
3.19].
(ii)
If
we
regard
Out
FC
(Π
n
)
as
a
closed
subgroup
of
Out
FC
(Π
1
)
by
means
of
the
natural
injection
Out
FC
(Π
n
)
→
Out
FC
(Π
1
)
of
[NodNon],
Theorem
B,
then
the
closed
subgroup
Dehn(G)
⊆
∼
(Aut(G)
⊆)
Out(Π
G
)
←
Out(Π
1
)
[cf.
[CbTpI],
Definition
4.4]
is
contained
in
Out
FC
(Π
n
)
geo
⊆
Out
FC
(Π
n
),
i.e.,
Dehn(G)
⊆
Out
FC
(Π
n
)
geo
.
COMBINATORIAL
ANABELIAN
TOPICS
II
97
Proof.
First,
we
verify
assertion
(i).
Let
us
observe
that
if
the
outer
isomorphism
φ
n
arises
scheme-theoretically
as
a
specialization
isomor-
phism
—
cf.
the
discussion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1
—
then
the
commutativity
in
question
follows
immediately
from
the
various
definitions
involved
[cf.
also
the
discus-
sion
preceding
[CmbCsp],
Definition
2.1].
Now
the
general
case
follows
from
the
observation
that
the
scheme-theoretic
case
treated
above
al-
lows
one
to
reduce
to
the
case
where
Y
log
=
X
log
,
and
φ
n
is
an
FC-
admissible
outomorphism,
in
which
case
the
commutativity
in
question
is
a
tautological
consequence
of
the
fact
that
T
Π
tpd
is
a
group
homomor-
phism.
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
The
inclusion
Dehn(G)
⊆
Out
FC
(Π
n
)
follows
immediately
from
the
fact
that
every
profinite
Dehn
multi-
twist
arises
scheme-theoretically.
Next,
we
observe
that
the
inclusion
Dehn(G)
⊆
Out
FC
(Π
n
)
geo
may
be
regarded
either
as
a
consequence
of
the
fact
that
every
profinite
Dehn
multi-twist
arises
“Q-scheme-
theoretically”,
i.e.,
from
scheme
theory
over
Q
[cf.
the
commutative
diagram
of
Remark
3.19.1],
or
as
a
consequence
of
the
following
argu-
ment:
Observe
that
it
follows
immediately
from
assertion
(i),
together
with
[CbTpI],
Theorem
4.8,
(ii),
(iv),
that,
by
applying
a
suitable
spe-
cialization
isomorphism
—
cf.
the
discussion
preceding
[CmbCsp],
Def-
inition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1
—
we
may
assume
with-
out
loss
of
generality
that
G
is
totally
degenerate.
Then
the
inclusion
Dehn(G)
⊆
Out
FC
(Π
n
)
geo
follows
immediately
from
Theorem
3.18,
(ii)
[cf.
also
Theorem
3.16,
(v);
[CbTpI],
Definition
4.4!].
This
completes
the
proof
of
assertion
(ii).
Corollary
3.25
(Co-Dehn-ness
of
degeneration
structures
in
the
totally
degenerate
case).
In
the
notation
of
Theorem
3.16,
for
i
=
1,
2,
let
Y
i
log
be
a
stable
log
curve
over
(Spec
k)
log
;
H
i
the
“G”
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
Y
i
log
;
(H
i
,
S
i
,
φ
i
)
a
3-
cuspidalizable
degeneration
structure
on
G
[cf.
Definition
3.23,
(i),
(v)];
α
i
∈
Out(Π
G
)
a
nondegenerate
(H
i
,
S
i
,
φ
i
)-Dehn
multi-twist
of
G
[cf.
Definition
3.23,
(iv)].
Suppose
that
α
1
commutes
with
α
2
,
and
that
H
2
is
totally
degenerate
[cf.
[CbTpI],
Definition
2.3,
(iv)].
Suppose,
moreover,
that
one
of
the
following
conditions
is
satisfied:
(a)
r
=
0.
(b)
α
1
and
α
2
are
positive
definite
[cf.
Definition
3.23,
(iv)].
Then
(H
1
,
S
1
,
φ
1
)
is
co-Dehn
to
(H
2
,
S
2
,
φ
2
)
[cf.
Definition
3.23,
(iii)],
or,
equivalently
[since
H
2
is
totally
degenerate],
(H
2
,
S
2
,
φ
2
)
(H
1
,
S
1
,
φ
1
)
[cf.
Definition
3.23,
(ii)].
98
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
Proof.
For
i
=
1,
2,
write
ψ
i
:
Π
G
→
Π
H
i
for
the
composite
outer
isomorphism
φ
i
∼
ψ
i
:
Π
G
←
Π
(H
i
)
Si
Φ
(H
i
)
S
i
∼
→
Π
H
i
and
ψ
=
ψ
1
◦
ψ
2
−1
.
Write
α
1
[H
2
]
∈
Out(Π
H
2
)
for
the
outomorphism
obtained
by
conjugating
α
1
by
ψ
2
.
First,
we
claim
that
the
following
assertion
holds:
def
Claim
3.25.A:
There
exists
a
positive
integer
a
such
def
that
β
=
α
1
[H
2
]
a
∈
Dehn(H
2
).
Indeed,
since
α
1
is
an
(H
1
,
S
1
,
φ
1
)-Dehn
multi-twist
of
G,
the
outomor-
phism
α
1
[H
2
]
of
Π
H
2
is
group-theoretically
cuspidal.
Thus,
since
α
1
commutes
with
α
2
,
it
follows,
in
the
case
of
condition
(a)
(respectively,
(b)),
from
Theorem
1.9,
(i)
(respectively
Theorem
1.9,
(ii)),
which
may
be
applied
in
light
of
[CbTpI],
Corollary
5.9,
(ii)
(respectively,
[CbTpI],
Corollary
5.9,
(iii)),
that
α
1
[H
2
]
∈
Aut(H
2
).
In
particular,
since
the
underlying
semi-graph
of
H
2
is
finite,
there
exists
a
positive
integer
a
such
that
α
1
[H
2
]
a
∈
Aut
|grph|
(H
2
)
[cf.
[CbTpI],
Definition
2.6,
(i);
Remark
4.1.2
of
the
present
monograph].
On
the
other
hand,
since
α
1
is
an
(H
1
,
S
1
,
φ
1
)-Dehn
multi-twist
of
G,
it
follows
immediately
from
Proposition
3.24,
(i),
(ii),
that
the
image
of
α
1
via
the
tripod
homo-
morphism
associated
to
Π
3
[cf.
Definition
3.19]
is
trivial.
Thus,
since
H
2
is
totally
degenerate,
and
α
1
[H
2
]
a
∈
Aut
|grph|
(H
2
),
by
applying
The-
orem
3.18,
(ii),
together
with
Proposition
3.24,
(i),
we
conclude
that
β
=
α
1
[H
2
]
a
∈
Dehn(H
2
).
This
completes
the
proof
of
Claim
3.25.A.
{l}
Next,
let
us
fix
an
element
l
∈
Σ.
For
i
∈
{1,
2},
write
H
i
for
the
semi-graph
of
anabelioids
of
pro-l
PSC-type
obtained
by
forming
the
pro-l
completion
of
H
i
[cf.
[SemiAn],
Definition
2.9,
(ii)].
Then
it
fol-
lows
immediately
from
Claim
3.25.A,
together
with
[CbTpI],
Theorem
4.8,
(ii),
(iv),
that
there
exists
a
subset
S
⊆
Node(H
2
)
[which
may
de-
{l}
pend
on
l!]
such
that
the
automorphism
β
{l}
∈
Aut(H
2
)
induced
by
β
{l}
{l}
{l}
is
contained
in
Dehn((H
2
)
S
)
⊆
Dehn(H
2
)
⊆
Aut(H
2
)
[i.e.,
β
{l}
is
{l}
a
profinite
Dehn
multi-twist
of
(H
2
)
S
],
and,
moreover,
β
{l}
is
nonde-
{l}
{l}
generate
as
a
profinite
Dehn
multi-twist
of
(H
2
)
S
.
Write
α
1
for
the
outomorphism
of
the
pro-l
group
Π
H
{l}
[which
is
naturally
isomorphic
1
to
the
maximal
pro-l
quotient
of
Π
H
1
]
obtained
by
conjugating
α
1
by
∼
ψ
1
and
ψ
{l}
:
Π
H
{l}
→
Π
H
{l}
for
the
outer
isomorphism
induced
by
ψ
2
1
[cf.
the
discussion
preceding
Claim
3.25.A].
Next,
we
claim
that
the
following
assertion
holds:
Claim
3.25.B:
The
composite
outer
isomorphism
ψ
S
:
Π
(H
2
)
S
Φ
(H
2
)
S
∼
→
ψ
∼
Π
H
2
→
Π
H
1
COMBINATORIAL
ANABELIAN
TOPICS
II
99
∼
is
graphic,
i.e.,
arises
from
an
isomorphism
(H
2
)
S
→
H
1
.
∼
Indeed,
let
ψ
S
:
Π
(H
2
)
S
→
Π
H
1
be
an
isomorphism
that
lifts
ψ
S
.
Then
it
follows
immediately
from
[CmbGC],
Proposition
1.5,
(ii)
—
by
con-
sidering
the
functorial
bijections
between
the
sets
“VCN”
[cf.
[NodNon],
Definition
1.1,
(iii)]
of
various
connected
finite
étale
coverings
of
H
1
,
(H
2
)
S
—
that,
to
verify
Claim
3.25.B,
it
suffices
to
verify
the
follow-
ing:
Let
I
2
→
(H
2
)
S
be
a
connected
finite
étale
cov-
ering
of
(H
2
)
S
that
corresponds
to
a
characteristic
open
subgroup
Π
I
2
⊆
Π
(H
2
)
S
.
Write
I
1
→
H
1
for
the
connected
finite
étale
covering
of
H
1
that
corre-
sponds
to
the
[necessarily
characteristic]
open
sub-
def
{l}
{l}
group
Π
I
1
=
ψ
S
(Π
I
2
)
⊆
Π
H
1
and
I
1
,
I
2
for
the
semi-graphs
of
anabelioids
of
pro-l
PSC-type
obtained
by
forming
the
pro-l
completions
of
I
1
,
I
2
,
respec-
∼
tively.
Then
the
outer
isomorphism
Π
I
{l}
→
Π
I
{l}
de-
2
1
termined
by
ψ
S
is
graphic.
To
verify
this
graphicity,
let
us
first
recall
that
the
automorphisms
{l}
β
{l}
∈
Aut((H
2
)
S
)
and
α
1
∈
Aut(H
1
)
are
nondegenerate
profinite
Dehn
multi-twists.
Thus,
it
follows
immediately
from
Lemma
3.26,
(i),
(ii),
below
[cf.
also
Claim
3.25.A],
that
there
exist
liftings
β
∈
1
∈
Aut(Π
H
1
)
of
β,
α
1
,
respectively,
and
a
positive
Aut(Π
(H
2
)
S
),
α
integer
b
such
that
the
outomorphisms
γ
2
,
γ
1
of
Π
I
{l}
,
Π
I
{l}
deter-
2
1
{l}
mined
by
β
b
,
α
b
are
nondegenerate
profinite
Dehn
multi-twists
of
I
,
{l}
1
2
I
1
,
respectively,
and,
moreover,
γ
2
and
γ
1
a
are
compatible
relative
to
∼
the
outer
isomorphism
in
question
Π
I
{l}
→
Π
I
{l}
.
Moreover,
if
condi-
2
1
tion
(b)
is
satisfied,
then
γ
1
is
a
positive
definite
profinite
Dehn
multi-
{l}
twist
of
I
1
[cf.
Lemma
3.26,
(ii),
below].
Thus,
it
follows,
in
the
case
of
condition
(a)
(respectively,
(b)),
from
Theorem
1.9,
(i)
(respec-
tively
Theorem
1.9,
(ii)),
which
may
be
applied
in
light
of
[CbTpI],
Corollary
5.9,
(ii)
(respectively,
[CbTpI],
Corollary
5.9,
(iii)),
that
the
∼
outer
isomorphism
in
question
Π
I
{l}
→
Π
I
{l}
is
graphic.
This
com-
2
1
pletes
the
proof
of
Claim
3.25.B.
On
the
other
hand,
one
verifies
eas-
ily
from
the
various
definitions
involved
that
Claim
3.25.B
implies
that
(H
2
,
S
2
,
φ
2
)
(H
1
,
S
1
,
φ
1
).
This
completes
the
proof
of
Corol-
lary
3.25.
Lemma
3.26
(Profinite
Dehn
multi-twists
and
pro-Σ
comple-
tions
of
finite
étale
coverings).
Let
Σ
1
⊆
Σ
0
be
nonempty
sets
of
prime
numbers,
G
0
a
semi-graph
of
anabelioids
of
pro-Σ
0
PSC-type,
100
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
H
0
→
G
0
a
connected
finite
étale
Galois
covering
that
arises
from
a
∈
Aut(Π
G
0
).
Write
normal
open
subgroup
Π
H
0
⊆
Π
G
0
of
Π
G
0
,
and
α
G
1
,
H
1
for
the
semi-graphs
of
anabelioids
of
pro-Σ
1
PSC-type
obtained
by
forming
the
pro-Σ
1
completions
of
G
0
,
H
0
,
respectively
[cf.
[SemiAn],
Definition
2.9,
(ii)].
Suppose
that
α
∈
Aut(Π
G
0
)
preserves
the
normal
open
subgroup
Π
H
0
⊆
Π
G
0
corresponding
to
H
0
→
G
0
.
Write
α
G
0
,
α
H
0
,
α
G
1
,
α
H
1
for
the
respective
outomorphisms
of
Π
G
0
,
Π
H
0
,
Π
G
1
,
Π
H
1
in-
duced
by
α
.
Suppose,
moreover,
that
α
G
0
∈
Dehn(G
0
)
[cf.
[CbTpI],
Definition
4.4].
Then
the
following
hold:
(i)
It
holds
that
α
G
1
∈
Dehn(G
1
).
Moreover,
there
exists
a
positive
integer
a
such
that
a
a
∈
Dehn(H
0
)
,
α
H
∈
Dehn(H
1
)
.
α
H
0
1
(ii)
If,
moreover,
α
G
1
∈
Dehn(G
1
)
[cf.
(i)]
is
nondegenerate
(re-
spectively,
positive
definite)
[cf.
[CbTpI],
Definition
5.8,
a
(ii),
(iii)],
then
α
H
∈
Dehn(H
1
)
[cf.
(i)]
is
nondegenerate
1
(respectively,
positive
definite).
Proof.
First,
we
verify
assertion
(i).
One
verifies
easily
from
[NodNon],
Lemma
2.6,
(i),
together
with
[CbTpI],
Corollary
5.9,
(i),
that
there
a
∈
Dehn(H
0
).
Now
since
exists
a
positive
integer
a
such
that
α
H
0
a
α
G
0
∈
Dehn(G
0
),
α
H
0
∈
Dehn(H
0
),
it
follows
immediately
from
the
a
∈
Dehn(H
1
).
various
definitions
involved
that
α
G
1
∈
Dehn(G
1
),
α
H
1
This
completes
the
proof
of
assertion
(i).
Assertion
(ii)
follows
imme-
diately,
in
the
nondegenerate
(respectively,
positive
definite)
case,
from
[NodNon],
Lemma
2.6,
(i),
together
with
[CbTpI],
Corollary
5.9,
(ii)
(respectively,
from
Corollary
5.9,
(iii),
(v)).
This
completes
the
proof
of
Lemma
3.26.
Corollary
3.27
(Commensurator
of
profinite
Dehn
multi-twists
in
the
totally
degenerate
case).
In
the
notation
of
Theorem
3.16,
Definition
3.19
[so
n
≥
3],
suppose
further
that
G
is
totally
degener-
def
ate
[cf.
[CbTpI],
Definition
2.3,
(iv)].
Write
s
:
Spec
k
→
(M
g,[r]
)
k
=
(M
g,[r]
)
Spec
k
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”]
for
the
underlying
(1-)morphism
of
algebraic
stacks
of
log
log
def
the
classifying
(1-)morphism
(Spec
k)
log
→
(M
g,[r]
)
k
=
(M
g,[r]
)
Spec
k
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”]
s
log
for
the
log
scheme
of
the
stable
log
curve
X
log
over
(Spec
k)
log
;
N
s
def
obtained
by
equipping
N
=
Spec
k
with
the
log
structure
induced,
via
log
s,
by
the
log
structure
of
(M
g,[r]
)
k
;
N
s
log
for
the
log
stack
obtained
by
log
by
the
nat-
forming
the
[stack-theoretic]
quotient
of
the
log
scheme
N
s
ural
action
of
the
finite
k-group
“s
×
(M
g,[r]
)
k
s”,
i.e.,
the
fiber
product
COMBINATORIAL
ANABELIAN
TOPICS
II
101
over
(M
g,[r]
)
k
of
two
copies
of
s;
N
s
for
the
underlying
stack
of
the
log
stack
N
s
log
;
I
N
s
⊆
π
1
(N
s
log
)
for
the
closed
subgroup
of
the
log
fundamen-
tal
group
π
1
(N
s
log
)
of
N
s
log
given
by
the
kernel
of
the
natural
surjection
π
1
(N
s
log
)
π
1
(N
s
)
[induced
by
the
(1-)morphism
N
s
log
→
N
s
obtained
(Σ)
by
forgetting
the
log
structure];
π
1
(N
s
log
)
for
the
quotient
of
π
1
(N
s
log
)
by
the
kernel
of
the
natural
surjection
from
I
N
s
to
its
maximal
pro-Σ
Σ
.
Then
the
following
hold:
quotient
I
N
s
(i)
The
natural
homomorphism
π
1
(N
s
log
)
→
Out(Π
1
)
[cf.
the
natu-
ral
outer
homomorphism
of
the
first
display
of
Remark
3.19.1]
(Σ)
factors
through
the
quotient
π
1
(N
s
log
)
π
1
(N
s
log
)
and
the
natural
inclusion
N
Out
FC
(Π
n
)
geo
(Dehn(G))
→
Out(Π
1
)
[cf.
Propo-
sition
3.24,
(ii)].
In
particular,
we
obtain
a
homomorphism
(Σ)
π
1
(N
s
log
)
−→
N
Out
FC
(Π
n
)
geo
(Dehn(G))
,
hence
also
a
homomorphism
(Σ)
π
1
(N
s
log
)
−→
C
Out
FC
(Π
n
)
geo
(Dehn(G))
.
(ii)
The
second
displayed
homomorphism
of
(i)
fits
into
a
natural
commutative
diagram
of
profinite
groups
1
−−−→
Σ
I
N
⏐
s
⏐
−−−→
(Σ)
π
1
(N
s
log
)
⏐
⏐
−−−→
π
1
(N
s
)
−−−→
1
⏐
⏐
1
−−−→
Dehn(G)
−−−→
C
Out
FC
(Π
n
)
geo
(Dehn(G))
−−−→
Aut(G)
−−−→
1
[cf.
Definition
3.1,
(ii),
concerning
the
notation
“G”]
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
isomorphisms.
(iii)
Dehn(G)
is
open
in
C
Out
FC
(Π
n
)
geo
(Dehn(G)).
(iv)
We
have
an
equality
N
Out
FC
(Π
n
)
geo
(Dehn(G))
=
C
Out
FC
(Π
n
)
geo
(Dehn(G))
.
Proof.
First,
we
verify
assertion
(i).
The
fact
that
the
image
of
the
homomorphism
in
question
is
contained
in
Out
FC
(Π
n
)
geo
follows
imme-
diately
from
the
[tautological!]
fact
that
this
image
arises
“Q-scheme-
theoretically”,
i.e.,
from
scheme
theory
over
Q
[cf.
the
discussion
of
Remark
3.19.1].
Thus,
assertion
(i)
follows
immediately
from
the
fact
that
the
natural
homomorphism
π
1
(N
s
log
)
→
Out(Π
1
)
determines
an
Σ
∼
isomorphism
I
N
→
Dehn(G)
[cf.
[CbTpI],
Proposition
5.6,
(ii)].
This
s
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
First,
let
us
observe
that
it
follows
from
[CbTpI],
Theorem
5.14,
(iii),
that
C
Out
FC
(Π
n
)
geo
(Dehn(G))
⊆
Aut(G).
Thus,
we
obtain
a
natural
homomorphism
C
Out
FC
(Π
n
)
geo
(Dehn(G))
→
Aut(G),
whose
kernel
contains
Dehn(G)
[cf.
the
definition
of
a
profinite
102
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Dehn
multi-twist
given
in
[CbTpI],
Definition
4.4].
On
the
other
hand,
if
an
element
α
∈
C
Out
FC
(Π
n
)
geo
(Dehn(G))
acts
trivially
on
G,
then,
since
G
is
totally
degenerate,
it
follows
immediately
from
Theorem
3.18,
(ii),
that
α
∈
Dehn(G).
This
completes
the
proof
of
the
existence
of
the
lower
exact
sequence
in
the
diagram
of
assertion
(ii),
except
for
the
surjectivity
of
the
third
arrow
of
this
sequence.
Thus,
it
follows
im-
mediately
from
the
proof
of
assertion
(i)
that,
to
complete
the
proof
of
assertion
(ii),
it
suffices
to
verify
that
the
right-hand
vertical
arrow
log
π
1
(N
s
)
→
Aut(G)
of
the
diagram
is
an
isomorphism.
Write
X
N
s
for
the
log
whose
classifying
(1-)morphism
is
given
by
stable
log
curve
over
N
s
log
→
(M
log
)
k
and
Aut
log
(X
log
)
for
the
the
natural
(1-)morphism
N
s
g,[r]
s
N
s
N
log
log
log
group
of
automorphisms
of
X
N
over
N
s
.
Then
since
X
,
hence
also
s
log
X
N
,
is
totally
degenerate,
one
verifies
easily
that
the
natural
homo-
s
log
morphism
Aut
N
s
log
(X
N
s
)
→
Aut(G)
is
an
isomorphism.
Thus,
it
follows
immediately
from
the
various
definitions
involved
that
the
right-hand
vertical
arrow
π
1
(N
s
)
→
Aut(G)
of
the
diagram
is
an
isomorphism.
This
completes
the
proof
of
assertion
(ii).
Assertion
(iii)
follows
immediately
from
the
exactness
of
the
lower
sequence
of
the
diagram
of
assertion
(ii),
together
with
the
finiteness
of
G.
Assertion
(iv)
follows
immediately
from
the
fact
that
the
middle
vertical
arrow
of
the
diagram
of
assertion
(ii)
is
an
isomorphism
which
factors
through
N
Out
FC
(Π
n
)
geo
(Dehn(G))
⊆
C
Out
FC
(Π
n
)
geo
(Dehn(G))
[cf.
assertion
(i)].
This
completes
the
proof
of
Corollary
3.27.
Remark
3.27.1.
One
interesting
consequence
of
Corollary
3.27
is
the
following:
The
profinite
group
Out
FC
(Π
n
)
geo
[which,
as
discussed
in
Remark
3.19.1,
may
be
regarded
as
the
geometric
portion
of
the
group
of
FC-admissible
outomorphisms
of
the
configuration
space
group
Π
n
],
hence
also
the
commensurator
C
Out
FC
(Π
n
)
geo
(Dehn(G)),
is
defined
in
a
purely
combinatorial/group-theoretic
fashion.
In
particular,
it
follows
from
the
commutative
diagram
of
Corollary
3.27,
(ii),
that
this
com-
mensurator
C
Out
FC
(Π
n
)
geo
(Dehn(G))
yields
a
purely
combinatorial/group-
theoretic
algorithm
for
reconstructing
the
profinite
groups
of
scheme-
theoretic
origin
that
appear
in
the
upper
sequence
of
this
diagram.
COMBINATORIAL
ANABELIAN
TOPICS
II
103
4.
Glueability
of
combinatorial
cuspidalizations
In
the
present
§4,
we
discuss
the
glueability
of
combinatorial
cuspidal-
izations.
The
resulting
theory
may
be
regarded
as
a
higher-dimensional
analogue
of
the
displayed
exact
sequence
of
[CbTpI],
Theorem
B,
(iii)
[cf.
Theorem
4.14,
(iii),
below,
of
the
present
monograph].
This
theory
implies
a
certain
key
surjectivity
property
of
the
tripod
homomorphism
[cf.
Corollary
4.15
below].
Finally,
we
apply
this
result
to
construct
cuspidalizations
of
the
log
fundamental
group
of
a
stable
log
curve
over
a
finite
field
[cf.
Corollary
4.16
below]
and
to
compute
certain
com-
mensurators
of
the
corresponding
Galois
image
in
the
totally
degenerate
case
[cf.
Corollary
4.17
below].
In
the
present
§4,
we
maintain
the
notation
of
the
preceding
§3
[cf.
also
Definition
3.1].
In
addition,
let
Σ
0
be
a
nonempty
set
of
prime
numbers
and
G
0
a
semi-graph
of
anabelioids
of
pro-Σ
0
PSC-type.
Write
G
0
for
the
underlying
semi-graph
of
G
0
and
Π
G
0
for
the
[pro-Σ
0
]
funda-
mental
group
of
G
0
.
Definition
4.1.
(i)
We
shall
write
Aut
|Brch(G
0
)|
(G
0
)
⊆
(Aut
|Vert(G
0
)|
(G
0
)
∩
Aut
|Node(G
0
)|
(G
0
)
⊆)
Aut(G
0
)
[cf.
[CbTpI],
Definition
2.6,
(i)]
for
the
[closed]
subgroup
of
Aut(G
0
)
consisting
of
automorphisms
α
of
G
0
that
induce
the
identity
automorphism
of
Vert(G
0
),
Node(G
0
)
and,
moreover,
fix
each
of
the
branches
of
every
node
of
G
0
.
Thus,
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
Aut
|grph|
(G
0
)
−→
Aut
|Brch(G
0
)|
(G
0
)
−→
Aut(Cusp(G
0
))
[cf.
[CbTpI],
Definition
2.6,
(i);
Remark
4.1.2
of
the
present
monograph].
(ii)
Let
v
∈
Vert(G
0
).
Then
we
shall
write
E(G
0
|
v
:
G
0
)
⊆
Edge(G
0
|
v
)
(=
Cusp(G
0
|
v
))
[cf.
[CbTpI],
Definition
2.1,
(iii)]
for
the
subset
of
Edge(G
0
|
v
)
(=
Cusp(G
0
|
v
))
consisting
of
cusps
of
G
0
|
v
that
arise
from
nodes
of
G
0
.
(iii)
We
shall
write
Aut
|E(G|
v
:G)|
(G
0
|
v
)
Glu
brch
(G
0
)
⊆
v∈Vert(G
0
)
[cf.
(ii);
[CbTpI],
Definition
2.6,
(i)]
for
the
[closed]
subgroup
of
|E(G|
v
:G)|
(G
0
|
v
)
consisting
of
“glueable”
collections
v∈Vert(G
0
)
Aut
104
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
of
automorphisms
of
the
various
G
0
|
v
,
i.e.,
the
subgroup
con-
sisting
of
(α
v
)
v∈Vert(G
0
)
such
that,
for
every
v,
w
∈
Vert(G
0
),
it
holds
that
χ
v
(α
v
)
=
χ
w
(α
w
)
[cf.
[CbTpI],
Definition
3.8,
(ii)].
Remark
4.1.1.
In
the
notation
of
Definition
4.1,
one
verifies
easily
from
the
various
definitions
involved
that
brch
|grph|
Glu(G
0
)
=
Glu
(G
0
)
∩
Aut
(G
0
|
v
)
v∈Vert(G
0
)
[cf.
[CbTpI],
Definitions
2.6,
(i),
and
4.9;
Remark
4.1.2
of
the
present
monograph].
Remark
4.1.2.
Here,
we
take
the
opportunity
to
correct
a
minor
error
in
the
exposition
of
[CbTpI].
In
[CbTpI],
Definition
2.6,
(i),
“Aut
|grph|
(G)”
should
be
defined
as
the
subgroup
of
Aut(G)
of
automor-
phisms
of
G
which
induce
the
identity
automorphism
on
the
underlying
semi-graph
of
G
[cf.
the
definition
given
in
[CbTpI],
Theorem
B].
In
a
similar
vein,
in
[CbTpI],
Definition
2.6,
(iii),
“Aut
|H|
(G)”
should
be
de-
fined
as
the
subgroup
of
Aut(G)
of
automorphisms
of
G
which
preserve
the
sub-semi-graph
H
of
the
underlying
semi-graph
of
G
and,
moreover,
induce
the
identity
automorphism
of
H.
Since
the
correct
definitions
are
applied
throughout
the
exposition
of
[CbTpI],
these
errors
in
the
statement
of
the
definitions
have
no
substantive
effect
on
the
exposition
of
[CbTpI],
except
for
the
following
two
instances
[which
themselves
do
not
have
any
substantive
effect
on
the
exposition
of
[CbTpI]]:
(i)
In
[CbTpI],
Proposition
2.7,
(ii),
“Aut
|grph|
(G)”
should
be
re-
placed
by
“Aut
|VCN(G)|
(G)”.
(ii)
In
[CbTpI],
Proposition
2.7,
(iii),
the
phrase
“In
particular”
should
be
replaced
by
the
word
“Finally”.
Theorem
4.2
(Glueability
of
combinatorial
cuspidalizations
in
the
one-dimensional
case).
Let
Σ
0
be
a
nonempty
set
of
prime
num-
bers
and
G
0
a
semi-graph
of
anabelioids
of
pro-Σ
0
PSC-type.
Write
Π
G
0
for
the
[pro-Σ
0
]
fundamental
group
of
G
0
.
Then
the
following
hold:
(i)
The
closed
subgroup
Dehn(G
0
)
⊆
Aut(G
0
)
[cf.
[CbTpI],
Def-
inition
4.4]
is
contained
in
Aut
|Brch(G
0
)|
(G
0
)
⊆
Aut(G
0
)
[cf.
Definition
4.1,
(i)],
i.e.,
Dehn(G
0
)
⊆
Aut
|Brch(G
0
)|
(G
0
).
(ii)
The
natural
homomorphism
Aut
|Brch(G
0
)|
(G
0
)
−→
α
→
v∈Vert(G
0
)
Aut(G
0
|
v
)
(α
G
0
|
v
)
v∈Vert(G
0
)
COMBINATORIAL
ANABELIAN
TOPICS
II
105
[cf.
[CbTpI],
Definition
2.14,
(ii);
[CbTpI],
Remark
2.5.1,
(ii)]
factors
through
Glu
brch
(G
0
)
⊆
Aut(G
0
|
v
)
v∈Vert(G
0
)
[cf.
Definition
4.1,
(iii)].
(iii)
The
natural
inclusion
Dehn(G
0
)
→
Aut
|Brch(G
0
)|
(G
0
)
of
(i)
and
|Brch(G
0
)|
the
natural
homomorphism
ρ
brch
(G
0
)
→
Glu
brch
(G
0
)
G
0
:
Aut
[cf.
(ii)]
fit
into
an
exact
sequence
of
profinite
groups
ρ
brch
G
0
1
−→
Dehn(G
0
)
−→
Aut
|Brch(G
0
)|
(G
0
)
−→
Glu
brch
(G
0
)
−→
1
.
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved.
Assertion
(ii)
follows
immediately
from
[CbTpI],
Corollary
3.9,
(iv).
Assertion
(iii)
follows,
in
light
of
Remark
4.1.1,
from
the
exact
sequence
of
[CbTpI],
Theorem
B,
(iii),
together
with
the
existence
of
automorphisms
of
G
0
that
induce
arbitrary
permutations
of
the
cusps
on
each
vertex
of
G
0
and,
moreover,
restrict
to
automorphisms
of
each
G
0
|
v
that
lie
in
the
kernel
of
χ
v
[cf.
the
automorphisms
constructed
in
the
proof
of
[CmbCsp],
Lemma
2.4].
Definition
4.3.
Let
H
be
a
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
of
G
[cf.
Definition
3.1,
(ii)]
and
S
⊆
Node(G|
H
)
[cf.
[CbTpI],
Definition
2.2,
(ii)]
a
subset
of
Node(G|
H
)
that
is
not
of
separating
type
[cf.
[CbTpI],
Definition
2.5,
(i)].
Then,
by
applying
a
similar
argument
to
the
argument
applied
in
[CmbCsp],
Definition
2.1,
(iii),
(vi),
or
[NodNon],
Definition
5.1,
(ix),
(x)
[i.e.,
by
considering
the
portion
of
the
underlying
scheme
X
n
of
X
n
log
corresponding
to
the
underlying
scheme
(X
H,S
)
n
of
the
n-th
log
configuration
space
(X
H,S
)
log
n
log
of
the
stable
log
curve
X
H,S
determined
by
(G|
H
)
S
—
cf.
[CbTpI],
Definition
2.5,
(ii)],
one
obtains
a
closed
subgroup
(Π
H,S
)
n
⊆
Π
n
[which
is
well-defined
up
to
Π
n
-conjugation].
We
shall
refer
to
(Π
H,S
)
n
⊆
Π
n
as
a
configuration
space
subgroup
[associated
to
(H,
S)].
For
each
0
≤
i
≤
j
≤
n,
we
shall
write
def
(Π
H,S
)
n/i
=
(Π
H,S
)
n
∩
Π
n/i
⊆
Π
n/i
[which
is
well-defined
up
to
Π
n
-conjugation];
def
(Π
H,S
)
j/i
=
(Π
H,S
)
n/i
/(Π
H,S
)
n/j
⊆
Π
j/i
[which
is
well-defined
up
to
Π
j
-conjugation].
In
particular,
(Π
H,S
)
j
=
(Π
H,S
)
j/0
⊆
Π
j
106
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[where
we
recall
that,
in
fact,
the
subgroups
on
either
side
of
the
“=”
are
only
well-defined
up
to
Π
j
-conjugation].
Thus,
by
applying
[CbTpI],
Proposition
2.11,
inductively,
we
conclude
that
each
(Π
H,S
)
j/i
is
a
pro-Σ
configuration
space
group
[cf.
[MzTa],
Definition
2.3,
(i)],
and
that
we
have
a
natural
exact
sequence
of
profinite
groups
1
−→
(Π
H,S
)
j/i
−→
(Π
H,S
)
j
−→
(Π
H,S
)
i
−→
1
.
Finally,
let
v
∈
Vert(G).
Then
the
semi-graph
of
anabelioids
of
PSC-
type
G|
v
[cf.
[CbTpI],
Definition
2.1,
(iii)]
may
be
naturally
identified
with
(G|
H
v
)
S
v
for
suitable
choices
of
H
v
,
S
v
[cf.
[CbTpI],
Remark
2.5.1,
(ii)].
We
shall
refer
to
def
(Π
v
)
n
=
(Π
H
v
,S
v
)
n
⊆
Π
n
as
a
configuration
space
subgroup
associated
to
v.
Thus,
(Π
v
)
1
⊆
Π
1
is
a
verticial
subgroup
associated
to
v
∈
Vert(G),
i.e.,
a
subgroup
that
is
typically
denoted
“Π
v
”.
We
shall
write
def
(Π
v
)
j/i
=
(Π
H
v
,S
v
)
j/i
⊆
Π
j/i
.
Remark
4.3.1.
In
the
notation
of
Definition
4.3,
one
verifies
easily
—
by
applying
a
suitable
specialization
isomorphism
[cf.
the
discus-
sion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1]
—
that
there
exist
a
stable
log
curve
Y
log
over
(Spec
k)
log
and
an
n-cuspidalizable
degeneration
structure
(G,
S,
φ)
on
Y
G
[cf.
Defini-
tion
3.23,
(i),
(v)]
—
where
we
write
Y
G
for
the
“G”
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
Y
log
—
which
satisfy
the
following:
Write
Y
Π
n
for
the
“Π
n
”
that
occurs
in
the
case
where
we
take
“X
log
”
to
be
Y
log
.
Then:
The
image
of
a
configuration
space
subgroup
of
Π
n
associated
to
(H,
S)
[cf.
Definition
4.3]
via
a
PFC-
∼
admissible
outer
isomorphism
Π
n
→
Y
Π
n
that
lies
over
the
displayed
composite
isomorphism
of
Definition
3.23,
(v)
[where
we
note
that,
in
loc.
cit.,
the
roles
of
“
Y
Π
n
”
and
“Π
n
”
are
reversed!],
is
a
configuration
space
sub-
group
of
Y
Π
n
associated
to
a
vertex
of
Y
G.
Lemma
4.4
(Commensurable
terminality
and
slimness).
Every
configuration
space
subgroup
[cf.
Definition
4.3]
of
Π
n
is
topo-
logically
finitely
generated,
slim,
and
commensurably
terminal
in
Π
n
.
Proof.
Since
any
configuration
space
subgroup
is,
in
particular,
a
con-
figuration
space
group,
the
fact
that
such
a
subgroup
is
topologically
finitely
generated
and
slim
follows
from
[MzTa],
Proposition
2.2,
(ii).
COMBINATORIAL
ANABELIAN
TOPICS
II
107
Thus,
it
remains
to
verify
commensurable
terminality.
By
applying
the
observation
of
Remark
4.3.1,
we
reduce
immediately
to
the
case
of
a
configuration
space
subgroup
associated
to
a
vertex.
But
then
the
desired
commensurable
terminality
follows,
in
light
of
Lemma
4.5
below,
by
induction
on
n,
together
with
the
corresponding
fact
for
n
=
1
[cf.
[CmbGC],
Proposition
1.2,
(ii)].
This
completes
the
proof
of
Lemma
4.4.
Lemma
4.5
(Extensions
and
commensurable
terminality).
Let
1
−−−→
N
H
−−−→
H
−−−→
Q
H
−−−→
1
⏐
⏐
⏐
⏐
⏐
⏐
1
−−−→
N
−−−→
G
−−−→
Q
−−−→
1
be
a
commutative
diagram
of
profinite
groups,
where
the
horizontal
se-
quences
are
exact,
and
the
vertical
arrows
are
injective.
Suppose
that
N
H
⊆
N
,
Q
H
⊆
Q
are
commensurably
terminal
in
N
,
Q,
respectively.
Then
H
⊆
G
is
commensurably
terminal
in
G.
Proof.
This
follows
immediately
from
Lemma
3.9,
(i).
Definition
4.6.
(i)
We
shall
write
Out
FC
(Π
n
)
brch
⊆
Out
FC
(Π
n
)
for
the
closed
subgroup
of
Out
FC
(Π
n
)
given
by
the
inverse
im-
age
of
∼
Aut
|Brch(G)|
(G)
⊆
(Aut(G)
⊆)
Out(Π
G
)
←
Out(Π
1
)
[cf.
Definition
4.1,
(i)]
via
the
natural
injection
Out
FC
(Π
n
)
→
Out
FC
(Π
1
)
⊆
Out(Π
1
)
of
[NodNon],
Theorem
B.
def
(ii)
Let
v
∈
Vert(G);
write
Π
v
=
(Π
v
)
1
[cf.
Definition
4.3].
Then
we
shall
write
Out
FC
((Π
v
)
n
)
G-node
⊆
Out
FC
((Π
v
)
n
)
for
the
[closed]
subgroup
of
Out
FC
((Π
v
)
n
)
given
by
the
inverse
image
of
Aut
|E(G|
v
:G)|
(G|
v
)
⊆
(Aut(G|
v
)
⊆)
Out(Π
v
)
[cf.
Definition
4.1,
(ii);
[CbTpI],
Definition
2.6,
(i)]
via
the
natural
injection
Out
FC
((Π
v
)
n
)
→
Out
FC
(Π
v
)
⊆
Out(Π
v
)
of
[NodNon],
Theorem
B.
108
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Theorem
4.7
(Graphicity
of
outomorphisms
of
certain
subquo-
tients).
In
the
notation
of
the
preceding
§3
[cf.
also
Definition
3.1],
let
x
∈
X
n
(k).
Write
C
x
⊆
Cusp(G)
for
the
[possibly
empty]
set
consisting
of
cusps
c
of
G
such
that,
for
some
i
∈
{1,
·
·
·
,
n},
x
{i}
∈
X
{i}
(k)
=
X(k)
[cf.
Definition
3.1,
(i)]
lies
on
the
cusp
of
X
log
corresponding
to
c
∈
Cusp(G).
For
each
i
∈
{1,
·
·
·
,
n},
write
def
G
i/i−1,x
=
G
i∈{1,···
,i},x
[cf.
Definition
3.1,
(iii)]
and
z
i/i−1,x
∈
VCN(G
i/i−1,x
)
for
the
element
of
VCN(G
i/i−1,x
)
on
which
x
{1,···
,i}
lies,
that
is
to
say:
If
x
{1,···
,i}
∈
X
i
(k)
[cf.
the
notation
given
in
the
discus-
sion
preceding
Definition
3.1]
is
a
cusp
or
node
of
the
log
log
→
X
i−1
geometric
fiber
of
the
projection
p
log
i/i−1
:
X
i
over
x
log
{1,···
,i−1}
corresponding
to
an
edge
e
∈
Edge(G
i/i−1,x
),
def
then
z
i/i−1,x
=
e;
if
x
{1,···
,i}
∈
X
i
(k)
is
neither
a
cusp
nor
node
of
the
geometric
fiber
of
the
projection
log
log
p
log
→
X
i−1
over
x
log
i/i−1
:
X
i
{1,···
,i−1}
but
lies
on
the
irre-
ducible
component
of
the
geometric
fiber
corresponding
def
to
a
vertex
v
∈
Edge(G
i/i−1,x
),
then
z
i/i−1,x
=
v.
Let
α
∈
Out
FC
(Π
n
)
brch
[cf.
Definition
4.6,
(i)].
Suppose
that
the
element
of
∼
Aut
|Brch(G)|
(G)
⊆
(Aut(G)
⊆)
Out(Π
G
)
←
Out(Π
1
)
[cf.
Definition
4.1,
(i)]
determined
by
α
∈
Out
FC
(Π
n
)
brch
[cf.
Defini-
tion
4.6,
(i)]
is
contained
in
Aut
|C
x
|
(G)
⊆
Aut(G)
[cf.
[CbTpI],
Definition
2.6,
(i)].
Then
there
exist
•
a
lifting
α
∈
Aut(Π
n
)
of
α,
and,
∼
•
for
each
i
∈
{1,
·
·
·
,
n},
a
VCN-subgroup
Π
z
i/i−1,x
⊆
Π
i/i−1
→
Π
G
i/i−1
,x
[cf.
Definition
3.1,
(iii)]
associated
to
the
element
z
i/i−1,x
∈
VCN(G
i/i−1,x
)
such
that
the
following
properties
hold:
∼
(a)
For
each
i
∈
{1,
·
·
·
,
n},
the
automorphism
of
Π
i/i−1
→
Π
G
i/i−1,x
∼
determined
by
α
fixes
the
VCN-subgroup
Π
z
i/i−1,x
⊆
Π
i/i−1
→
Π
G
i/i−1,x
.
COMBINATORIAL
ANABELIAN
TOPICS
II
109
∼
(b)
For
each
i
∈
{1,
·
·
·
,
n},
the
outomorphism
of
Π
i/i−1
→
Π
G
i/i−1,x
induced
by
α
is
contained
in
∼
Aut
|Brch(G
i/i−1,x
)|
(G
i/i−1,x
)
⊆
Out(Π
G
i/i−1,x
)
←
Out(Π
i/i−1
)
.
Proof.
We
verify
Theorem
4.7
by
induction
on
n.
If
n
=
1,
then
Theo-
rem
4.7
follows
immediately
from
the
various
definitions
involved.
Now
suppose
that
n
≥
2,
and
that
the
induction
hypothesis
is
in
force.
In
particular,
[since
the
homomorphism
p
Π
n/n−1
:
Π
n
Π
n−1
is
surjective]
we
have
a
lifting
α
∈
Aut(Π
n
)
of
α
and,
for
each
i
∈
{1,
·
·
·
,
n
−
1},
∼
a
VCN-subgroup
Π
z
i/i−1,x
⊆
Π
i/i−1
→
Π
G
i/i−1
,x
associated
to
the
ele-
ment
z
i/i−1,x
∈
VCN(G
i/i−1,x
)
such
that,
for
each
i
∈
{1,
·
·
·
,
n
−
1},
the
automorphism
of
Π
i
determined
by
α
fixes
Π
z
i/i−1,x
⊆
Π
i/i−1
⊆
Π
i
,
and,
moreover,
the
automorphism
of
Π
n−1
determined
by
α
satisfies
the
property
(b)
in
the
statement
of
Theorem
4.7.
Now
we
claim
that
the
following
assertion
holds:
∼
Claim
4.7.A:
The
outomorphism
of
Π
n/n−1
→
Π
G
n/n−1,x
induced
by
the
lifting
α
is
contained
in
∼
Aut
|Brch(G
n/n−1,x
)|
(G
n/n−1,x
)
⊆
Out(Π
G
n/n−1,x
)
←
Out(Π
n/n−1
)
.
To
this
end,
let
us
first
observe
that
it
follows
immediately
—
by
re-
log
log
→
X
n−2
via
a
suit-
placing
X
n
log
by
the
base-change
of
p
log
n/n−2
:
X
n
log
able
morphism
of
log
schemes
(Spec
k)
log
→
X
n−2
whose
image
lies
on
x
{1,···
,n−2}
∈
X
n−2
(k)
—
from
Lemma
3.2,
(iv),
that,
to
verify
Claim
4.7.A,
we
may
assume
without
loss
of
generality
that
n
=
2.
Also,
one
verifies
easily,
by
applying
Lemma
3.14,
(i)
[cf.
also
[CbTpI],
Proposi-
tion
2.9,
(i)],
and
possibly
replacing,
when
z
1/0,x
∈
Vert(G
1/0,x
),
•
α
by
the
composite
of
α
with
an
inner
automorphism
of
Π
n
=
Π
2
determined
by
conjugation
by
a
suitable
element
of
Π
n
=
∼
Π
2
whose
image
in
Π
1
→
Π
G
1/0,x
is
contained
in
the
closed
∼
subgroup
Π
z
1/0,x
⊆
Π
G
1/0,x
←
Π
1
and
•
x
by
a
suitable
“x”
whose
associated
“z
1/0,x
”
is
a
node
of
G
1/0,x
that
abuts
to
the
original
z
1/0,x
∈
Vert(G
1/0,x
),
that
we
may
assume
without
loss
of
generality
that
z
1/0,x
∈
Edge(G
1/0,x
).
∼
Next,
let
us
recall
that
the
automorphism
of
Π
1
→
Π
G
1/0,x
determined
∼
by
α
fixes
the
edge-like
subgroup
Π
z
1/0,x
⊆
Π
1
→
Π
G
1/0,x
associated
to
the
edge
z
1/0,x
of
G
1/0,x
[cf.
the
discussion
preceding
Claim
4.7.A].
Thus,
since
[we
have
assumed
that]
α
∈
Out
FC
(Π
2
)
brch
[which
implies
∼
that
the
outomorphism
of
Π
1
→
Π
G
1/0,x
determined
by
α
preserves
the
∼
Π
1
-conjugacy
class
of
each
verticial
subgroup
of
Π
1
→
Π
G
1/0,x
],
it
fol-
lows
immediately
from
Lemma
3.13,
(i),
(ii),
that
the
outomorphism
∼
of
Π
G
2/1,x
←
Π
2/1
induced
by
α
is
group-theoretically
verticial,
hence
[cf.
[NodNon],
Proposition
1.13;
[CmbGC],
Proposition
1.5,
(ii);
the
fact
that
α
is
C-admissible]
graphic,
i.e.,
∈
Aut(G
2/1,x
).
Moreover,
since
110
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
the
outomorphism
of
Π
G
2∈{2},x
←
Π
1
induced
by
α
is,
by
assumption,
|Brch(G)|
(G)
[cf.
[CmbCsp],
Proposition
1.2,
(iii)],
one
contained
in
Aut
verifies
easily,
by
considering
the
map
on
vertices/nodes/branches
in-
duced
by
the
projection
p
Π
{1,2}/{2}
|
Π
2/1
:
Π
2/1
Π
{2}
∼
[cf.
Lemma
3.6,
(i),
(iv)],
that
the
outomorphism
of
Π
G
2/1,x
←
Π
2/1
induced
by
α
is
contained
in
the
subgroup
Aut
|Brch(G
2/1,x
)|
(G
2/1,x
).
This
completes
the
proof
of
Claim
4.7.A.
On
the
other
hand,
one
verifies
easily
from
Claim
4.7.A,
together
with
the
various
definitions
involved,
that
there
exist
a
Π
n/n−1
-conjugate
β
∼
of
α
and
a
VCN-subgroup
Π
z
n/n−1
,x
⊆
Π
n/n−1
→
Π
G
n/n−1,x
associated
to
z
n/n−1,x
∈
VCN(G
n/n−1,x
)
such
that
β
fixes
Π
z
n/n−1
,x
.
In
particular,
the
lifting
β
of
α
and
the
VCN-subgroups
Π
z
i/i−1
,x
[where
i
∈
{1,
·
·
·
,
n}]
satisfy
the
properties
(a),
(b)
in
the
statement
of
Theorem
4.7.
This
completes
the
proof
of
Theorem
4.7.
Lemma
4.8
(Preservation
of
configuration
space
subgroups).
The
following
hold:
(i)
Let
α
∈
Out
FC
(Π
n
)
brch
[cf.
Definition
4.6,
(i)].
Then
α
pre-
serves
the
Π
n
-conjugacy
class
of
each
configuration
space
sub-
group
[cf.
Definition
4.3]
of
Π
n
.
Thus,
by
applying
the
portion
of
Lemma
4.4
concerning
commensurable
terminality,
to-
gether
with
Lemma
3.10,
(i),
we
obtain
a
natural
homomor-
phism
Out
FC
(Π
n
)
brch
−→
Out((Π
v
)
n
)
.
v∈Vert(G)
(ii)
The
displayed
homomorphism
of
(i)
factors
through
Out
FC
((Π
v
)
n
)
G-node
⊆
v∈Vert(G)
Out((Π
v
)
n
)
v∈Vert(G)
[cf.
Definition
4.6,
(ii)].
Proof.
First,
we
verify
assertion
(i).
We
begin
by
observing
that,
in
light
of
the
observation
of
Remark
4.3.1
[cf.
also
[CbTpI],
Proposition
2.9,
(ii)],
to
complete
the
verification
of
assertion
(i),
it
suffices
to
verify
the
following
assertion:
Claim
4.8.A:
For
each
v
∈
Vert(G),
α
preserves
the
Π
n
-conjugacy
class
of
configuration
space
subgroups
(Π
v
)
n
⊆
Π
n
of
Π
n
associated
to
v.
COMBINATORIAL
ANABELIAN
TOPICS
II
111
To
verify
Claim
4.8.A,
let
us
observe
that,
by
applying
Theorem
4.7
in
the
case
where
we
take
the
“x”
in
the
statement
of
Theorem
4.7
to
be
such
that,
for
each
i
∈
{1,
·
·
·
,
n},
the
element
z
i/i−1,x
∈
Vert(G
i/i−1,x
)
is
the
vertex
of
G
i/i−1,x
that
corresponds
[via
the
various
bijections
of
Lemma
3.6,
(iii)]
to
the
vertex
v
of
Claim
4.8.A,
we
obtain,
for
each
∼
i
∈
{1,
·
·
·
,
n},
a
VCN-subgroup
Π
z
i/i−1,x
⊆
Π
i/i−1
→
Π
G
i/i−1
,x
associ-
ated
to
z
i/i−1,x
∈
VCN(G
i/i−1,x
)
as
in
the
statement
of
Theorem
4.7,
(a).
Next,
let
us
observe
that
one
verifies
immediately
from
the
com-
mensurable
terminality
[cf.
[CmbGC],
Proposition
1.2,
(ii)]
of
each
of
the
VCN-subgroups
Π
z
i/i−1,x
⊆
Π
i/i−1
,
where
i
∈
{1,
·
·
·
,
n},
that
the
Π
n
-conjugacy
class
of
the
configuration
space
subgroup
(Π
v
)
n
⊆
Π
n
coincides
with
the
Π
n
-conjugacy
class
of
the
closed
subgroup
of
Π
n
consisting
of
γ
∈
Π
n
such
that,
for
each
i
∈
{1,
·
·
·
,
n},
conjuga-
tion
by
γ
preserves
the
closed
subgroup
Π
z
i/i−1,x
⊆
(Π
i/i−1
⊆)
Π
i
[so
Π
z
i/i−1,x
=
(Π
v
)
i/i−1
].
Thus,
it
follows
from
Theorem
4.7,
(a),
that
α
preserves
the
Π
n
-conjugacy
class
of
(Π
v
)
n
⊆
Π
n
,
as
desired.
This
completes
the
proof
of
Claim
4.8.A.
Next,
we
verify
assertion
(ii).
Let
α
∈
Out
FC
(Π
n
)
brch
,
v
∈
Vert(G).
Write
α
v
for
the
outomorphism
of
(Π
v
)
n
induced
by
α
[cf.
(i)].
Then
the
F-admissibility
of
α
v
follows
immediately
from
the
F-admissibility
of
α
[cf.
the
discussion
of
Definition
4.3].
The
C-admissibility
of
α
v
follows
immediately
from
Theorem
4.7
[applied
as
in
the
proof
of
Claim
4.8.A];
[NodNon],
Lemma
1.7,
together
with
the
definition
of
C-admissibility.
Finally,
the
fact
that
α
v
∈
Out
FC
((Π
v
)
n
)
G-node
follows
immediately
from
the
fact
that
α
∈
Out
FC
(Π
n
)
brch
.
This
completes
the
proof
of
assertion
(ii).
Definition
4.9.
We
shall
write
Out
FC
((Π
v
)
n
)
G-node
Glu(Π
n
)
⊆
v∈Vert(G)
for
the
[closed]
subgroup
of
v∈Vert(G)
Out
FC
((Π
v
)
n
)
G-node
consisting
of
“glueable”
collections
of
outomorphisms
of
the
various
(Π
v
)
n
,
i.e.,
the
subgroup
defined
as
follows:
(i)
Suppose
that
n
=
1.
Then
Glu(Π
n
)
consists
of
those
collections
(α
v
)
v∈Vert(G)
such
that,
for
every
v,
w
∈
Vert(G),
it
holds
that
χ
v
(α
v
)
=
χ
w
(α
w
)
[cf.
[CbTpI],
Definition
3.8,
(ii)]
—
where
we
note
that
one
verifies
easily
that
α
v
may
be
regarded
as
an
element
of
Aut(G|
v
).
(ii)
Suppose
that
n
=
2.
Then
Glu(Π
n
)
consists
of
those
collections
(α
v
)
v∈Vert(G)
that
satisfy
the
following
condition:
Let
v,
w
∈
Vert(G);
e
∈
N
(v)
∩
N
(w);
T
⊆
Π
2/1
⊆
Π
2
=
Π
n
a
{1,
2}-
tripod
of
Π
n
arising
from
e
∈
N
(v)
∩
N
(w)
[cf.
Definitions
112
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
3.3,
(i);
3.7,
(i)].
Then
one
verifies
easily
from
the
various
definitions
involved
that
there
exist
Π
n
-conjugates
T
v
,
T
w
of
T
such
that
T
v
,
T
w
are
contained
in
(Π
v
)
n
,
(Π
w
)
n
,
respectively,
and,
moreover,
T
v
⊆
(Π
v
)
2/1
⊆
(Π
v
)
2
=
(Π
v
)
n
,
T
w
⊆
(Π
w
)
2/1
⊆
(Π
w
)
2
=
(Π
w
)
n
are
tripods
of
(Π
v
)
n
,
(Π
w
)
n
arising
from
[the
cusps
of
G|
v
,
G|
w
corresponding
to]
the
node
e,
respectively.
Moreover,
since
α
v
∈
Out
FC
((Π
v
)
n
)
G-node
,
α
w
∈
Out
FC
((Π
w
)
n
)
G-node
,
it
follows
from
Theorem
3.16,
(iv),
that
α
v
∈
Out
FC
((Π
v
)
n
)[T
v
],
α
w
∈
∼
Out
FC
((Π
w
)
n
)[T
w
];
thus,
we
obtain
that
T
T
v
(α
v
)
∈
Out(T
v
)
→
∼
Out(T
);
T
T
w
(α
w
)
∈
Out(T
w
)
→
Out(T
)
[cf.
Theorem
3.16,
(i)].
Then
we
require
that
T
T
v
(α
v
)
=
T
T
w
(α
w
).
(iii)
Suppose
that
n
≥
3.
Then
Glu(Π
n
)
consists
of
those
col-
lections
(α
v
)
v∈Vert(G)
that
satisfy
the
following
condition:
Let
Π
tpd
⊆
Π
3
be
a
3-central
{1,
2,
3}-tripod
of
Π
n
[cf.
Definitions
3.3,
(i);
3.7,
(ii)].
Then
one
verifies
easily
from
the
various
definitions
involved
that,
for
every
v
∈
Vert(G),
there
exists
a
of
Π
tpd
such
that
Π
tpd
is
contained
in
(Π
v
)
3
,
Π
3
-conjugate
Π
tpd
v
v
tpd
and,
moreover,
Π
v
⊆
(Π
v
)
3
is
a
3-central
tripod
of
(Π
v
)
3
.
(α
v
)
∈
Thus,
since
α
v
∈
Out
FC
((Π
v
)
n
)
G-node
,
we
obtain
T
Π
tpd
v
tpd
∼
tpd
Out(Π
v
)
→
Out(Π
)
[cf.
Theorem
3.16,
(i),
(v)].
Then,
for
(α
v
)
=
T
Π
tpd
(α
w
).
every
v,
w
∈
Vert(G),
we
require
that
T
Π
tpd
v
w
Remark
4.9.1.
In
the
notation
of
Definition
4.9,
one
verifies
eas-
ily
from
the
various
definitions
involved
that
the
natural
outer
iso-
∼
∼
moprhism
Π
1
→
Π
G
determines
a
natural
isomorphism
Glu(Π
1
)
→
Glu
brch
(G)
[cf.
Definition
4.1,
(iii)].
Lemma
4.10
(Basic
properties
concerning
groups
of
glueable
collections).
For
n
≥
1,
the
following
hold:
(i)
The
natural
injections
Out
FC
((Π
v
)
n+1
)
→
Out
FC
((Π
v
)
n
)
of
[NodNon],
Theorem
B
—
where
v
ranges
over
the
vertices
of
G
—
determine
an
injection
Glu(Π
n+1
)
→
Glu(Π
n
)
.
COMBINATORIAL
ANABELIAN
TOPICS
II
113
(ii)
The
displayed
homomorphism
of
Lemma
4.8,
(i),
Out
FC
(Π
n
)
brch
−→
Out((Π
v
)
n
)
v∈Vert(G)
factors
through
Glu(Π
n
)
⊆
Out((Π
v
)
n
)
.
v∈Vert(G)
Proof.
First,
we
verify
assertion
(i).
The
fact
that
the
image
of
the
composite
Glu(Π
n+1
)
→
Out
FC
((Π
v
)
n+1
)
→
v∈Vert(G)
Out
FC
((Π
v
)
n
)
v∈Vert(G)
is
contained
in
Out
FC
((Π
v
)
n
)
G-node
⊆
v∈Vert(G)
Out
FC
((Π
v
)
n
)
v∈Vert(G)
follows
immediately
from
the
various
definitions
involved.
The
fact
that
the
image
of
the
composite
Glu(Π
n+1
)
→
Out
FC
((Π
v
)
n+1
)
→
v∈Vert(G)
Out
FC
((Π
v
)
n
)
v∈Vert(G)
is
contained
in
Out
FC
((Π
v
)
n
)
G-node
Glu(Π
n
)
⊆
v∈Vert(G)
follows
immediately
from
the
various
definitions
involved
when
n
≥
3
and
from
Theorems
3.16,
(iv),
(v);
3.18,
(ii)
[applied
to
each
(Π
v
)
n+1
!],
when
n
=
2.
Thus,
it
remains
to
verify
assertion
(i)
in
the
case
where
n
=
1.
Suppose
that
n
=
1.
Let
(α
v
)
v∈Vert(G)
∈
Glu(Π
2
).
Write
((α
v
)
1
)
v∈Vert(G)
∈
v∈Vert(G)
Out
FC
((Π
v
)
1
)
G-node
for
the
image
of
(α
v
)
v∈Vert(G)
.
Since
G
is
connected,
to
verify
assertion
(i)
in
the
case
where
n
=
1,
it
suffices
to
verify
that,
for
any
two
vertices
v,
w
of
G
such
that
N
(v)
∩
N
(w)
=
∅,
it
holds
that
χ
v
((α
v
)
1
)
=
χ
w
((α
w
)
1
).
Let
x
∈
X
2
(k)
be
a
k-valued
geometric
point
of
X
2
such
that
x
{1}
∈
X(k)
[cf.
Definition
3.1,
(i)]
is
a
node
of
X
log
corresponding
to
an
element
of
N
(v)
∩
N
(w)
=
∅.
Then
by
applying
Theorem
4.7
to
a
suitable
lifting
α
v
(∈
Aut
FC
((Π
v
)
2
))
of
the
outomorphism
α
v
of
(Π
v
)
2
[where
we
take
the
“Π
n
”
in
the
statement
of
Theorem
4.7
to
be
(Π
v
)
2
],
we
∼
conclude
that
the
outomorphism
(α
v
)
2/1
of
Π
(G|
v
)
2∈{1,2},x
←
(Π
v
)
2/1
[cf.
Definition
3.1,
(iii)]
determined
by
α
v
is
graphic
and
fixes
each
of
the
vertices
of
(G|
v
)
2∈{1,2},x
.
Thus,
if
we
write
(α
v
)
{2}
for
the
outomorphism
of
the
“Π
{2}
”
that
occurs
in
the
case
where
we
take
“Π
2
”
to
be
(Π
v
)
2
,
then
it
follows
from
[CmbCsp],
Proposition
1.2,
(iii),
together
with
the
C-admissibility
of
(α
v
)
1
,
that
(α
v
)
{2}
is
C-admissible,
i.e.,
∈
Aut(G|
v
).
114
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Now,
for
a
[{1,
2}-]tripod
T
v
⊆
(Π
v
)
2
arising
from
the
cusp
x
{1}
of
G|
v
[cf.
Definitions
3.3,
(i);
3.7,
(i)],
we
compute:
χ
G|
v
((α
v
)
{2}
)
[cf.
[CmbCsp],
Proposition
1.2,
(iii)]
χ
G|
v
((α
v
)
1
)
=
=
χ
(G|
v
)
2∈{1,2},x
((α
v
)
2/1
)
[cf.
[CbTpI],
Corollary
3.9,
(iv)]
[cf.
[CbTpI],
Corollary
3.9,
(iv)]
=
χ
T
v
((α
v
)
2/1
|
T
v
)
[where
we
refer
to
Lemma
3.12,
(i),
concerning
“(α
v
)
2/1
|
T
v
”,
and
we
write
χ
T
v
for
the
“χ”
associated
to
the
vertex
of
(G|
v
)
2∈{1,2},x
corre-
sponding
to
T
v
].
Moreover,
by
applying
a
similar
argument
to
the
above
argument,
we
conclude
that
there
exists
a
lifting
α
w
of
α
w
such
that
∼
w
the
outomorphism
(α
w
)
2/1
of
Π
(G|
w
)
2∈{1,2},x
←
(Π
w
)
2/1
determined
by
α
is
graphic
[and
fixes
each
of
the
vertices
of
(G|
w
)
2∈{1,2},x
],
and,
more-
over,
for
a
[{1,
2}-]tripod
T
w
⊆
(Π
w
)
2
arising
from
the
cusp
x
{1}
of
G|
w
,
it
holds
that
χ
G|
w
((α
w
)
1
)
=
χ
T
w
((α
w
)
2/1
|
T
w
).
On
the
other
hand,
since
(α
v
)
v∈Vert(G)
∈
Glu(Π
2
),
it
holds
that
χ
T
v
((α
v
)
2/1
|
T
v
)
=
χ
T
w
((α
w
)
2/1
|
T
w
).
In
particular,
we
obtain
that
χ
G|
v
((α
v
)
1
)
=
χ
G|
w
((α
w
)
1
).
This
completes
the
proof
of
assertion
(i).
Next,
we
verify
assertion
(ii).
If
n
=
1,
then
assertion
(ii)
amounts
to
Theorem
4.2,
(ii)
[cf.
also
Remark
4.9.1].
If
n
≥
2,
then
assertion
(ii)
follows
immediately
from
Lemma
4.8,
(ii),
together
with
the
fact
that
the
homomorphism
“T
T
”
of
Theorem
3.16,
(i),
does
not
depend
on
the
choice
of
“T
”
among
its
conjugates.
This
completes
the
proof
of
assertion
(ii).
Definition
4.11.
We
shall
write
ρ
brch
for
the
homomorphism
n
Out
FC
(Π
n
)
brch
−→
Glu(Π
n
)
determined
by
the
factorization
of
Lemma
4.10,
(ii).
Lemma
4.12
(Glueable
collections
in
the
case
of
precisely
one
node).
Suppose
that
n
=
2,
and
that
#Node(G)
=
1.
Let
v
,
w
∈
Vert(
G)
be
distinct
elements
such
that
N
(
v
)
∩
N
(
w)
=
∅.
Write
e
∈
Node(
G)
for
the
unique
element
of
N
(
v
)∩N
(
w)
[cf.
[NodNon],
Lemma
∼
∼
1.8];
Π
v
,
Π
w
,
Π
e
⊆
Π
G
←
Π
1
for
the
VCN-subgroups
of
Π
G
←
Π
1
as-
def
respectively;
v
def
sociated
to
v
,
w,
e
∈
VCN(
G),
=
v
(G);
w
=
w(G);
def
e
=
e
(G).
[Thus,
one
verifies
easily
that
Π
e
=
Π
v
∩
Π
w
[cf.
[NodNon],
Lemma
1.9,
(i)],
that
Vert(G)
=
{v,
w},
and
that
if
G
is
noncycli-
cally
primitive
(respectively,
cyclically
primitive)
[cf.
[CbTpI],
Definition
4.1],
then
v
=
w
(respectively,
v
=
w).]
Let
x
∈
X
2
(k)
be
a
k-valued
geometric
point
of
X
2
such
that
x
{1}
∈
X(k)
[cf.
Defini-
tion
3.1,
(i)]
lies
on
the
unique
node
of
X
log
[i.e.,
which
corresponds
def
to
e].
Write
G
2/1
=
G
2∈{1,2},x
[cf.
Definition
3.1,
(iii)];
G
2/1
→
G
2/1
COMBINATORIAL
ANABELIAN
TOPICS
II
115
∼
for
the
profinite
étale
covering
corresponding
to
Π
G
2/1
←
Π
2/1
;
v
new
new
for
the
“v
2,1,x
”
of
Lemma
3.6,
(iv).
For
each
z
∈
Vert(G),
write
◦
z
∈
Vert(G
2/1
)
for
the
vertex
of
G
2/1
that
corresponds
to
z
via
the
bijections
of
Lemma
3.6,
(i),
(iv).
[Thus,
it
follows
from
Lemma
3.6,
(iv),
that
Vert(G
2/1
)
=
{v
new
,
v
◦
,
w
◦
}.]
Then
the
following
hold
[cf.
also
Figures
2,
3,
below]:
(i)
Let
(Π
v
)
2
⊆
Π
2
be
a
configuration
space
subgroup
of
Π
2
as-
sociated
to
v
[cf.
Definition
4.3]
such
that
the
image
of
the
p
Π
2/1
∼
composite
(Π
v
)
2
→
Π
2
Π
1
coincides
with
Π
v
⊆
Π
G
←
Π
1
.
∼
Also,
let
us
fix
a
verticial
subgroup
Π
v
new
⊆
Π
G
2/1
←
Π
2/1
of
∼
Π
G
2/1
←
Π
2/1
associated
to
a
v
new
∈
Vert(
G
2/1
)
that
lies
over
v
new
∈
Vert(G
2/1
)
and
is
contained
in
(Π
v
)
2
.
Then
there
exists
a
unique
configuration
space
subgroup
(Π
w
)
2
⊆
Π
2
of
Π
2
associated
to
w
[cf.
Definition
4.3]
such
that
Π
v
new
=
def
(Π
v
)
2/1
∩
(Π
w
)
2/1
—
where
we
write
(Π
v
)
2/1
=
Π
2/1
∩
(Π
v
)
2
;
def
(Π
w
)
2/1
=
Π
2/1
∩
(Π
w
)
2
—
and,
moreover,
the
image
of
the
p
Π
2/1
composite
(Π
w
)
2
→
Π
2
Π
1
coincides
with
Π
w
⊆
Π
1
.
(ii)
In
the
situation
of
(i),
the
natural
homomorphism
lim
(Π
v
←
Π
e
→
Π
w
)
−→
Π
1
−→
—
where
the
inductive
limit
is
taken
in
the
category
of
pro-
Σ
groups
—
is
injective,
and
its
image
is
commensurably
terminal
in
Π
1
.
Write
Π
v
,
w
⊆
Π
1
for
the
image
of
the
above
p
Π
2/1
homomorphism;
Π
2
|
Π
v
,
w
(⊆
Π
2
)
for
the
fiber
product
of
Π
2
Π
1
and
Π
v
,
w
→
Π
1
.
Thus,
we
have
an
exact
sequence
of
profinite
groups
1
−→
Π
2/1
−→
Π
2
|
Π
v
,
w
−→
Π
v
,
w
−→
1
.
Finally,
if
G
is
noncyclically
primitive,
then
Π
v
,
w
=
Π
1
,
Π
2
|
Π
v
,
w
=
Π
2
.
∼
(iii)
In
the
situation
of
(ii),
for
each
z
∈
{
v
,
w},
let
Π
z
◦
⊆
Π
G
2/1
←
∼
Π
2/1
be
a
verticial
subgroup
of
Π
G
2/1
←
Π
2/1
associated
to
a
z
◦
∈
Vert(
G
2/1
)
that
lies
over
z
◦
∈
Vert(G
2/1
)
such
that
Π
z
◦
⊆
(Π
z
)
2/1
[cf.
(i)],
and,
moreover,
Π
z
◦
∩
Π
v
new
=
{1}.
Thus,
∼
def
Π
e
z
◦
=
Π
z
◦
∩
Π
v
new
is
the
nodal
subgroup
of
Π
G
2/1
←
Π
2/1
associated
to
the
unique
element
e
z
◦
of
N
(
z
◦
)
∩
N
(
v
new
)
[cf.
def
[NodNon],
Lemma
1.9,
(i)].
Write
e
z
◦
=
e
z
◦
(G
2/1
).
Then
the
natural
homomorphism
lim
(Π
z
◦
←
Π
e
z
◦
→
Π
v
new
)
−→
(Π
z
)
2/1
−→
116
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
G
2/1
Π
H
v
◦
Π
H
w
◦
||
||
(Π
v
)
2/1
(Π
w
)
2/1
e
v
◦
e
w
◦
···
•
•
•
···
◦
new
·
·
v
·
v
w
◦
·
·
·
·
·
·
·
·
·
∨
•
v
◦
↓
·
·
·
·
·
·
·
·
∨
•
•
new
e
v
◦
v
e
w
◦
w
◦
G
=
G
1
···
•
v
Π
v
,
w
•
···
e
w
↓
•
v
e
•
w
Figure
2
:
the
noncyclically
primitive
case
COMBINATORIAL
ANABELIAN
TOPICS
II
G
2/1
Π
H
v
◦
117
Π
H
w
◦
||
||
(Π
v
)
2/1
(Π
w
)
2/1
e
v
◦
e
w
◦
•
•
•
◦
·
new
·
v
·
v
w
◦
·
···
·
·
·
·
·
·
·
·
·
·
·
·
·
·
e
v
◦
·
>
G
=
G
1
···
•
v
·
·
·
·
↓
·
·
·
·
new
·
·
v
·
·
•
·
·
<
·
•
◦
v
=
w
◦
e
w
◦
Π
v
,
w
•
e
w
···
↓
e
•
v
=
w
Figure
3
:
the
cyclically
primitive
case
···
118
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
—
where
the
inductive
limit
is
taken
in
the
category
of
pro-
Σ
groups
—
is
an
isomorphism.
Write
G
†
z
◦
for
the
sub-
semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
of
the
underlying
semi-graph
of
G
2/1
whose
set
of
vertices
=
def
{
z
(G)
◦
,
v
new
};
T
z
◦
=
(Node(G
2/1
)
\
{e
z
◦
})
∩
Node(G
2/1
|
G
†
◦
)
⊆
z
Node(G
2/1
)
[cf.
[CbTpI],
Definition
2.2,
(ii)].
Then
the
natu-
ral
homomorphism
of
the
above
display
allows
one
to
identify
(Π
z
)
2/1
with
the
[pro-Σ]
fundamental
group
Π
H
z
◦
of
def
H
z
◦
=
(G
2/1
|
G
†
◦
)
T
z
◦
z
[cf.
[CbTpI],
Definition
2.5,
(ii)].
(iv)
In
the
situation
of
(iii),
let
(α
z
)
z∈Vert(G)
∈
Glu(Π
2
).
Write
((α
z
)
1
)
z∈Vert(G)
∈
Glu(Π
1
)
for
the
image
of
(α
z
)
z∈Vert(G)
∈
Glu(Π
2
)
via
the
injection
of
Lemma
4.10,
(i).
Let
α
1
∈
Aut
|Brch(G)|
(G)
be
such
that
ρ
brch
(α
1
)
=
((α
z
)
1
)
z∈Vert(G)
∈
Glu(Π
1
)
[cf.
Theo-
1
rem
4.2,
(iii);
Definition
4.11].
Then
the
outomorphism
α
1
of
Π
1
preserves
the
Π
1
-conjugacy
class
of
Π
v
,
w
⊆
Π
1
.
Thus,
by
applying
the
portion
of
(ii)
concerning
commensurable
termi-
nality,
we
obtain
[cf.
Lemma
3.10,
(i)]
a
restricted
outomor-
phism
α
1
|
Π
v
,
w
∈
Out(Π
v
,
w
).
(v)
In
the
situation
of
(iv),
there
exists
an
outomorphism
β
v
,
w
[α
1
]
of
Π
2
|
Π
v
,
w
that
satisfies
the
following
conditions:
(1)
β
v
,
w
[α
1
]
preserves
Π
2/1
⊆
Π
2
|
Π
v
,
w
and
the
Π
2
|
Π
v
,
w
-conjugacy
classes
of
(Π
v
)
2
,
(Π
w
)
2
⊆
Π
2
|
Π
v
,
w
.
(2)
There
exists
an
automorphism
β
v
,
w
[α
1
]
of
Π
2
|
Π
v
,
w
that
lifts
the
outomorphism
β
v
,
w
[α
1
]
such
that
the
outomorphism
of
∼
Π
G
2/1
←
Π
2/1
determined
by
β
v
,
w
[α
1
]
[cf.
(1)]
is
con-
∼
tained
in
Aut
|Brch(G
2/1
)|
(G
2/1
)
⊆
Out(Π
G
2/1
)
←
Out(Π
2/1
).
(3)
For
each
z
∈
{
v
,
w},
the
outomorphism
β
v
,
w
[α
1
]|
(Π
z
)
2
of
(Π
z
)
2
determined
by
β
v
,
w
[α
1
]
[i.e.,
obtained
by
applying
(1)
and
Lemma
3.10,
(i)
—
where
we
note
that
(Π
z
)
2
is
commensurably
terminal
in
Π
2
[cf.
Lemma
4.4],
hence
also
in
Π
2
|
Π
v
,
w
]
coincides
with
α
z
(G)
[cf.
the
notation
of
(iv)].
(4)
The
outomorphism
of
Π
v
,
w
induced
by
β
v
,
w
[α
1
]
[cf.
(1)]
coincides
with
α
1
|
Π
v
,
w
[cf.
(iv)].
Here,
we
observe,
in
the
context
of
(2),
that
the
outer
isomor-
∼
phism
Π
2/1
→
Π
G
2/1
[i.e.,
which
gives
rise
to
“the”
closed
sub-
∼
group
Aut
|Brch(G
2/1
)|
(G
2/1
)
⊆
Out(Π
G
2/1
)
←
Out(Π
2/1
)]
may
be
characterized,
up
to
composition
with
elements
of
the
subgroup
COMBINATORIAL
ANABELIAN
TOPICS
II
119
∼
Aut
|Brch(G
2/1
)|
(G
2/1
)
⊆
Out(Π
G
2/1
)
←
Out(Π
2/1
),
as
the
group-
theoretically
cuspidal
[cf.
[CmbGC],
Definition
1.4,
(iv)]
outer
isomorphism
such
that
the
semi-graph
of
anabelioids
structure
on
G
2/1
is
the
semi-graph
of
anabelioids
structure
determined
[cf.
[NodNon],
Theorem
A]
by
the
resulting
composite
∼
∼
Π
e
→
Π
G
←
Π
1
→
Out(Π
2/1
)
→
Out(Π
G
2/1
)
—
where
the
third
arrow
is
the
outer
action
determined
by
the
p
Π
2/1
exact
sequence
1
→
Π
2/1
→
Π
2
→
Π
1
→
1
—
in
a
fashion
compatible
with
the
projection
p
Π
{1,2}/{2}
|
Π
2/1
:
Π
2/1
Π
{2}
and
∼
∼
the
given
outer
isomorphisms
Π
{2}
→
Π
1
→
Π
G
.
Proof.
First,
we
verify
assertion
(i).
The
existence
of
such
a
(Π
w
)
2
⊆
Π
2
follows
immediately
from
the
various
definitions
involved.
Thus,
it
remains
to
verify
the
uniqueness
of
such
a
(Π
w
)
2
.
Let
(Π
w
)
2
⊆
Π
2
be
as
in
assertion
(i)
and
γ
∈
Π
2
an
element
such
that
the
conjugate
(Π
w
)
γ
2
of
(Π
w
)
2
by
γ
satisfies
the
condition
on
“(Π
w
)
2
”
stated
in
assertion
(i).
Then
since
Π
w
is
commensurably
terminal
in
Π
1
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
it
holds
that
the
image
of
γ
via
p
Π
2/1
is
contained
in
Π
w
.
Thus
—
by
multiplying
γ
by
a
suitable
element
of
(Π
w
)
2
—
we
may
assume
without
loss
of
generality
that
γ
∈
Π
2/1
.
In
particular,
since
Π
v
new
⊆
(Π
w
)
2/1
∩(Π
w
)
γ
2/1
—
where
we
write
(Π
w
)
γ
2/1
=
Π
2/1
∩(Π
w
)
γ
2
—
is
not
abelian
[cf.
[CmbGC],
Remark
1.1.3],
it
follows
immediately
from
[NodNon],
Lemma
1.9,
(i),
that
(Π
w
)
2/1
=
(Π
w
)
γ
2/1
.
Thus,
since
(Π
w
)
2/1
is
commensurably
terminal
in
Π
2/1
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
it
holds
that
γ
∈
(Π
w
)
2/1
.
This
completes
the
proof
of
assertion
(i).
Assertions
(ii),
(iii),
(iv)
follow
immediately
from
the
various
def-
initions
involved
[cf.
also
[CmbGC],
Propositions
1.2,
(ii),
and
1.5,
(i),
as
well
as
the
proofs
of
[CmbCsp],
Proposition
1.5,
(iii);
[CbTpI],
Proposition
2.11].
Finally,
we
verify
assertion
(v).
It
follows
immediately
from
the
def-
inition
of
“Out
FC
((Π
(−)
)
2
)
G-node
”
[cf.
Definitions
4.6,
(ii);
4.9]
that,
for
each
z
∈
{
v
,
w},
there
exists
a
lifting
α
z
∈
Aut((Π
z
)
2
)
of
α
z
(G)
such
that
z
,
then
if
we
write
(
α
z
)
1
for
the
automorphism
of
Π
z
determined
by
α
(
α
z
)
1
(Π
e
)
=
Π
e
.
Next,
let
us
observe
that
it
follows
immediately
from
assertion
(ii)
that
the
automorphisms
(
α
v
)
1
,
(
α
w
)
1
[i.e.,
determined
w
]
determine
an
automorphism
α
1
|
Π
v
,
w
of
Π
v
,
w
.
by
the
liftings
α
v
,
α
Moreover,
let
us
also
observe
that
it
follows
immediately
from
Theo-
rem
4.2,
(iii)
[cf.
also
the
definition
of
profinite
Dehn
multi-twists
given
in
[CbTpI],
Definition
4.4],
that
the
assignment
“α
1
→
α
1
|
Π
v
,
w
”
implicit
in
assertion
(iv)
is
injective.
Thus,
one
verifies
immediately
from
the
definition
of
profinite
Dehn
multi-twists
that
one
may
choose
the
re-
w
of
α
v
,
α
w
so
that
(
α
v
)
1
(Π
e
)
=
(
α
w
)
1
(Π
e
)
=
Π
e
,
spective
liftings
α
v
,
α
def
120
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
and,
moreover,
the
outomorphism
of
Π
v
,
w
determined
by
the
result-
ing
automorphism
α
1
|
Π
v
,
w
coincides
with
the
outomorphism
α
1
|
Π
v
,
w
of
assertion
(iv).
Now
we
claim
that
the
following
assertion
holds:
Claim
4.12.A:
Write
(
α
z
)
2/1
for
the
automorphism
of
z
and
(α
z
)
2/1
for
the
outo-
(Π
z
)
2/1
determined
by
α
α
z
)
2/1
.
Then
—
morphism
of
(Π
z
)
2/1
determined
by
(
∼
relative
to
the
natural
identification
Π
H
z
◦
→
(Π
z
)
2/1
of
assertion
(iii)
—
it
holds
that
∈
Aut
|Brch(H
z
◦
)|
(H
z
◦
)
∼
→
Out((Π
z
)
2/1
))
.
(⊆
Out(Π
H
z
◦
)
Indeed,
careful
inspection
of
the
various
definitions
involved
reveals
that
Claim
4.12.A
follows
immediately
from
Theorem
4.7
[together
with
the
commensurable
terminality
of
the
subgroup
Π
e
⊆
Π
z
—
cf.
[CmbGC],
Proposition
1.2,
(ii)].
Thus
—
by
replacing
α
z
by
the
com-
posite
of
α
z
with
an
inner
automorphism
determined
by
conjugation
by
a
suitable
element
of
(Π
z
)
2/1
—
we
may
assume
without
loss
of
generality
that
α
z
(Π
e
z
◦
)
=
Π
e
z
◦
.
Moreover,
since
[cf.
Claim
4.12.A]
α
z
preserves
the
(Π
z
)
2/1
-conjugacy
classes
of
Π
z
◦
and
Π
v
new
,
and
the
∼
verticial
subgroups
Π
z
◦
,
Π
v
new
⊆
Π
G
2/1
←
Π
2/1
are
the
unique
verti-
∼
cial
subgroups
of
Π
G
2/1
←
Π
2/1
associated
to
z
(G)
◦
,
v
new
∈
Vert(G
2/1
),
respectively,
such
that
Π
e
z
◦
⊆
Π
z
◦
,
Π
e
z
◦
⊆
Π
v
new
[cf.
[CmbGC],
Propo-
sition
1.5,
(i)],
we
thus
conclude
that
α
z
(Π
z
◦
)
=
Π
z
◦
,
α
z
(Π
v
new
)
=
Π
v
new
.
Next,
write
(α
z
)
z
◦
,
(α
z
)
v
new
for
the
respective
outomorphisms
of
Π
z
◦
,
Π
v
new
determined
by
α
z
.
Now
we
claim
that
the
following
assertion
holds:
Claim
4.12.B:
It
holds
that
(α
z
)
2/1
(α
v
)
v
new
=
(α
w
)
v
new
.
Moreover,
if
v
=
w,
i.e.,
G
is
cyclically
primitive,
then
∼
—
relative
to
the
natural
outer
isomorphism
Π
v
◦
→
Π
w
◦
[where
we
note
that
if
v
=
w,
then
Π
v
◦
is
a
Π
2/1
-
conjugate
of
Π
w
◦
]
—
it
holds
that
(α
v
)
v
◦
=
(α
w
)
w
◦
.
Indeed,
the
equality
(α
v
)
v
new
=
(α
w
)
v
new
follows
from
the
definition
of
Glu(Π
2
).
Next,
suppose
that
G
is
cyclically
primitive.
To
verify
the
equality
(α
v
)
v
◦
=
(α
w
)
w
◦
,
let
us
observe
that,
for
each
z
∈
{
v
,
w},
the
p
Π
{1,2}/{2}
∼
composite
Π
z
◦
→
Π
2
Π
{2}
→
Π
G
is
injective
[and
its
image
is
a
verticial
subgroup
of
Π
G
associated
to
z
(G)
∈
Vert(G)].
Thus,
to
verify
the
equality
(α
v
)
v
◦
=
(α
w
)
w
◦
,
it
suffices
to
verify
that
the
outomor-
phism
of
the
image
of
Π
v
◦
in
Π
{2}
induced
by
(α
v
)
v
◦
coincides
with
the
outomorphism
of
the
image
of
Π
w
◦
in
Π
{2}
induced
by
(α
w
)
w
◦
.
On
the
COMBINATORIAL
ANABELIAN
TOPICS
II
121
other
hand,
this
follows
immediately
from
the
fact
that
both
α
v
and
α
w
are
liftings
of
the
same
outomorphism
α
v
=
α
w
of
“(Π
v
)
2
”=“(Π
w
)
2
”
[cf.
[CmbCsp],
Proposition
1.2,
(iii)].
This
completes
the
proof
of
Claim
4.12.B.
Next,
let
us
observe
that
it
follows
immediately
from
the
various
definitions
involved
that
if
G
is
noncyclically
primitive
(respectively,
cyclically
primitive),
then
#Vert((G
2/1
)
{e
v
◦
}
)
=
2
(respectively,
=
1),
and
that,
relative
to
the
correspondence
discussed
in
[CbTpI],
Propo-
sition
2.9,
(i),
(3),
H
v
◦
and
G
2/1
|
w
◦
(G)
(respectively,
H
v
◦
)
correspond(s)
to
the
two
vertices
(respectively,
the
unique
vertex)
of
(G
2/1
)
{e
v
◦
}
.
Next,
let
us
observe
the
following
equalities
[cf.
the
notation
of
[CbTpI],
Definition
3.8,
(ii)]:
χ
H
v
◦
((α
v
)
2/1
)
=
=
=
=
χ
H
z
◦
|
v
new
((α
v
)
v
new
)
χ
H
v
◦
|
v
new
((α
w
)
v
new
)
χ
H
w
◦
((α
w
)
2/1
)
χ
G
2/1
|
w
◦
(G)
((α
w
)
w
◦
)
[cf.
[CbTpI],
Corollary
3.9,
(iv)]
[cf.
Claim
4.12.B]
[cf.
[CbTpI],
Corollary
3.9,
(iv)]
[cf.
[CbTpI],
Corollary
3.9,
(iv)].
Now
it
follows
immediately
from
these
equalities,
together
with
Claim
4.12.A,
that
the
data
((α
v
)
2/1
,
(α
w
)
w
◦
)
∈
Aut(H
v
◦
)
×
Aut(G
2/1
|
w
◦
(G)
)
(respectively,
(α
v
)
2/1
∈
Aut(H
v
◦
)
)
may
be
regarded
as
an
element
of
Glu
brch
((G
2/1
)
{e
v
◦
}
)
[cf.
Defini-
tion
4.1,
(iii)].
Thus,
by
applying
the
exact
sequence
of
Theorem
4.2,
(iii)
[cf.
also
Remark
4.9.1],
we
conclude
that
there
exists
an
element
v
]
∈
Aut
|Brch((G
2/1
)
{ev
◦
}
)|
((G
2/1
)
{e
v
◦
}
)
α
2/1
[
of
a
collection
of
outomorphisms
of
Φ
(G
2/1
)
{ev
◦
}
∼
−→
Π
(G
2/1
)
{e
◦
}
v
∼
Π
G
2/1
−→
Π
2/1
[i.e.,
contained
in
the
image
of
Aut((G
2/1
)
{e
v
◦
}
)
→
Out(Π
2/1
)
—
cf.
[CbTpI],
Definition
2.10]
that
admits
a
natural
structure
of
torsor
over
Dehn((G
2/1
)
{e
v
◦
}
)
(⊆
Aut
|Brch((G
2/1
)
{ev
◦
}
)|
((G
2/1
)
{e
v
◦
}
)).
A
similar
argument
yields
the
existence
of
an
element
∈
Aut
|Brch((G
2/1
)
{ew
◦
}
)|
((G
2/1
)
{e
w
◦
}
)
α
2/1
[
w]
of
a
collection
of
outomorphisms
of
Φ
(G
Π
(G
2/1
)
{e
◦
}
w
2/1
)
{ew
◦
}
∼
−→
∼
Π
G
2/1
−→
Π
2/1
[i.e.,
contained
in
the
image
of
Aut((G
2/1
)
{e
w
◦
}
)
→
Out(Π
2/1
)]
that
admits
a
natural
structure
of
torsor
over
Dehn((G
2/1
)
{e
w
◦
}
)
(⊆
Aut
|Brch((G
2/1
)
{ew
◦
}
)|
((G
2/1
)
{e
w
◦
}
)).
122
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Now
we
claim
that
the
following
assertion
holds:
Claim
4.12.C:
For
each
z
∈
{
v
,
w},
the
automorphism
(
α
z
)
1
of
Π
z
is
compatible
with
the
outomorphism
α
2/1
[
z
]
of
Π
2/1
relative
to
the
homomorphism
Π
z
→
Π
1
→
Out(Π
2/1
)
—
where
the
second
arrow
is
the
natural
outer
action
determined
by
the
exact
sequence
p
Π
2/1
1
−→
Π
2/1
−→
Π
2
−→
Π
1
−→
1
.
v
],
it
follows
im-
Indeed,
to
verify
the
compatibility
of
(
α
v
)
1
and
α
2/1
[
mediately
from
the
various
definitions
involved
that
it
suffices
to
verify
def
α
v
)
1
(σ)
∈
Π
v
,
then
there
exist
that,
for
each
σ
∈
Π
v
,
if
we
write
τ
=
(
liftings
σ
,
τ
∈
Π
2
of
σ,
τ
∈
Π
v
,
respectively,
such
that
the
equality
[which
is
in
fact
independent
of
the
choice
of
liftings]
v
]
◦
[Inn(
σ
)]
◦
α
2/1
[
v
]
−1
=
[Inn(
τ
)]
∈
Out(Π
2/1
)
α
2/1
[
—
where
we
write
“Inn(−)”
for
the
automorphism
of
Π
2/1
determined
by
conjugation
by
“(−)”
and
“[Inn(−)]”
for
the
outomorphism
of
Π
2/1
determined
by
this
automorphism
—
holds.
To
this
end,
let
σ
∈
(Π
v
)
2
be
a
lifting
of
σ
∈
Π
v
.
Then
since
(Π
v
)
2/1
⊆
(Π
v
)
2
is
normal,
Inn(
σ
)
preserves
(Π
v
)
2/1
.
Next,
let
us
observe
that
the
semi-graph
of
anabelioids
structure
of
(G
2/1
)
{e
v
◦
}
[with
respect
to
which
w
◦
is
a
vertex
if
G
is
noncyclically
primitive
and,
moreover,
with
respect
to
which
e
w
◦
is
a
node
in
both
the
cyclically
primitive
and
noncyclically
primitive
cases]
may
be
thought
of
as
the
semi-graph
of
anabelioids
structure
on
the
fiber
subgroup
Π
2/1
[cf.
Definition
3.1,
(iii)]
arising
from
a
point
of
X
log
that
lies
in
the
1-interior
of
the
irreducible
component
of
X
log
corresponding
to
v.
Now
it
follows
immediately
from
this
observation
that
Inn(
σ
)
preserves
the
Π
2/1
-conjugacy
class
of
Π
w
◦
,
as
well
as
the
Π
2/1
-conjugacy
class
of
Π
e
w
◦
=
(Π
v
)
2/1
∩
Π
w
◦
if
G
is
noncyclically
primitive
(respectively,
pre-
serves
the
Π
2/1
-conjugacy
class
of
Π
e
w
◦
if
G
is
cyclically
primitive).
By
considering
the
various
Π
2/1
-conjugates
of
Π
e
w
◦
and
Π
w
◦
and
applying
[CmbGC],
Propositions
1.2,
(ii);
1.5,
(i),
we
thus
conclude
that
Inn(
σ
)
preserves
the
(Π
v
)
2/1
-conjugacy
classes
of
Π
e
w
◦
,
Π
w
◦
if
G
is
noncycli-
cally
primitive
(respectively,
preserves
the
(Π
v
)
2/1
-conjugacy
class
of
Π
e
w
◦
if
G
is
cyclically
primitive).
In
particular
—
by
multiplying
σ
by
a
suitable
element
of
(Π
v
)
2/1
—
we
may
assume
without
loss
of
gener-
ality
that
Inn(
σ
)
preserves
(Π
v
)
2/1
,
Π
w
◦
,
and
Π
e
w
◦
in
the
noncyclically
primitive
case
(respectively,
preserves
(Π
v
)
2/1
and
Π
e
w
◦
in
the
cyclically
primitive
case).
Next,
let
us
observe
that
one
verifies
easily
[cf.
Lemma
3.6,
(iv)]
that
p
Π
{1,2}/{2}
the
composite
Π
e
w
◦
→
Π
2
Π
{2}
surjects
onto
a
nodal
subgroup
∼
of
Π
G
←
Π
{2}
associated
to
e
∈
Node(G).
Thus,
since
Inn(
σ
)
preserves
Π
e
w
◦
,
it
follows
[cf.
[CmbGC],
Proposition
1.2,
(ii)]
that
the
image
of
COMBINATORIAL
ANABELIAN
TOPICS
II
σ
∈
Π
2
via
Π
2
p
Π
{1,2}/{2}
123
Π
{2}
is
contained
in
the
image
of
the
composite
p
Π
{1,2}/{2}
Π
e
w
◦
→
Π
2
Π
{2}
.
In
particular
—
by
multiplying
σ
by
a
suitable
element
of
Π
e
w
◦
(⊆
(Π
v
)
2/1
)
—
we
may
assume
without
loss
of
generality
that
σ
∈
Ker(p
Π
{1,2}/{2}
).
A
similar
argument
implies
that
α
v
)
1
(σ)
∈
Π
v
such
that
Inn(
τ
)
there
exists
a
lifting
τ
∈
(Π
v
)
2
of
τ
=
(
preserves
(Π
v
)
2/1
,
Π
w
◦
,
Π
e
w
◦
if
G
is
noncyclically
primitive
(respectively,
preserves
(Π
v
)
2/1
and
Π
e
w
◦
if
G
is
cyclically
primitive),
and,
moreover,
τ
∈
Ker(p
Π
{1,2}/{2}
).
α
v
)
1
of
(Π
v
)
2/1
,
Π
v
,
respec-
Now
since
the
automorphisms
(
α
v
)
2/1
,
(
tively,
arise
from
the
automorphism
α
v
of
(Π
v
)
2
,
it
follows
immediately
v
]
that
the
equality
from
the
construction
of
α
2/1
[
v
]
◦
[Inn(
σ
)]
◦
α
2/1
[
v
]
−1
=
[Inn(
τ
)]
α
2/1
[
holds
upon
restriction
to
[an
equality
of
outomorphisms
of]
(Π
v
)
2/1
.
Moreover,
if
G
is
noncyclically
primitive,
then
since
the
composite
p
Π
{1,2}/{2}
Π
w
◦
→
Π
2
Π
{2}
is
injective
[and
its
image
is
a
verticial
sub-
∼
group
of
Π
G
←
Π
{2}
associated
to
w
∈
Vert(G)
—
cf.
Lemma
3.6,
(iv)],
to
verify
the
restriction
of
the
equality
v
]
◦
[Inn(
σ
)]
◦
α
2/1
[
v
]
−1
=
[Inn(
τ
)]
α
2/1
[
to
[an
equality
of
outomorphisms
of]
Π
w
◦
,
it
suffices
to
verify
that
the
outomorphism
of
the
image
of
Π
w
◦
in
Π
{2}
induced
by
the
product
v
]
◦
[Inn(
σ
)]
◦
α
2/1
[
v
]
−1
◦
[Inn(
τ
)]
−1
α
2/1
[
is
trivial.
On
the
other
hand,
this
follows
immediately
from
the
fact
that
σ
,
τ
∈
Ker(p
Π
{1,2}/{2}
).
Thus,
in
summary,
the
restriction
of
the
equality
in
question
[i.e.,
in
the
discussion
immediately
following
Claim
4.12.C]
to
[an
equality
of
outomorphisms
of]
(Π
v
)
2/1
holds.
Moreover,
if
G
is
noncyclically
prim-
itive,
then
the
restriction
of
the
equality
in
question
to
[an
equality
of
outomorphisms
of]
Π
w
◦
holds.
In
particular,
it
follows
immediately
from
the
displayed
exact
sequence
of
Theorem
4.2,
(iii)
[cf.
also
Re-
mark
4.9.1],
that
the
product
α
2/1
[
v
]
◦
[Inn(
σ
)]
◦
α
2/1
[
v
]
−1
◦
[Inn(
τ
)]
−1
is
contained
in
Dehn((G
2/1
)
{e
v
◦
}
).
Thus
—
by
considering
the
outo-
morphism
of
Π
{2}
induced
by
the
above
product
—
one
verifies
eas-
ily
from
[CbTpI],
Theorem
4.8,
(iv),
together
with
the
fact
that
σ
,
Π
τ
∈
Ker(p
{1,2}/{2}
),
that
the
equality
in
question
holds.
This
completes
v
].
The
compatibility
of
the
proof
of
the
compatibility
of
(
α
v
)
1
and
α
2/1
[
follows
from
a
similar
argument.
This
completes
the
(
α
w
)
1
and
α
2/1
[
w]
proof
of
Claim
4.12.C.
Next,
we
claim
that
the
following
assertion
holds:
124
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Claim
4.12.D:
The
difference
α
2/1
[
v
]◦α
2/1
[
w]
−1
∈
Out(Π
2/1
)
∼
is
contained
in
Dehn(G
2/1
)
(⊆
Out(Π
G
2/1
)
←
Out(Π
2/1
)).
Indeed,
this
follows
immediately
from
the
two
displayed
equalities
of
Claim
4.12.B,
together
with
the
construction
of
α
2/1
[
v
],
α
2/1
[
w].
This
completes
the
proof
of
Claim
4.12.D.
Thus,
it
follows
immediately
from
Claim
4.12.D,
together
with
the
existence
of
the
natural
isomorphism
∼
Dehn((G
2/1
)
{e
v
◦
}
)
⊕
Dehn((G
2/1
)
{e
w
◦
}
)
−→
Dehn(G
2/1
)
[cf.
[CbTpI],
Theorem
4.8,
(ii),
(iv)],
that
—
by
replacing
α
2/1
[
v
],
α
2/1
[
w]
by
the
composites
of
α
2/1
[
v
],
α
2/1
[
w]
with
suitable
elements
of
Dehn((G
2/1
)
{e
v
◦
}
),
Dehn((G
2/1
)
{e
w
◦
}
),
respectively
[where
we
re-
v
],
α
2/1
[
w]
belong
to
torsors
over
call
that
the
outomorphisms
α
2/1
[
Dehn((G
2/1
)
{e
v
◦
}
),
Dehn((G
2/1
)
{e
w
◦
}
),
respectively]
—
we
may
as-
sume
without
loss
of
generality
that
α
2/1
[
v
]
=
α
2/1
[
w]
.
def
v
]
=
α
2/1
[
w].
Then
it
follows
immediately
from
Write
β
2/1
=
α
2/1
[
Claim
4.12.C,
together
with
the
fact
that
Π
v
,
w
is
topologically
gener-
ated
by
Π
v
,
Π
w
⊆
Π
v
,
w
[cf.
assertion
(ii)],
that
the
outomorphism
β
2/1
of
Π
2/1
is
compatible
with
the
automorphism
α
1
|
Π
v
,
w
of
Π
v
,
w
[i.e.,
the
α
w
)
1
—
cf.
the
discussion
immedi-
automorphism
induced
by
(
α
v
)
1
,
(
ately
preceding
Claim
4.12.A],
relative
to
the
composite
Π
v
,
w
→
Π
1
→
Out(Π
2/1
),
where
the
second
arrow
is
the
outer
action
determined
by
the
displayed
exact
sequence
of
Claim
4.12.C.
In
particular,
by
con-
∼
out
sidering
the
natural
isomorphism
Π
2
|
Π
v
,
w
→
Π
2/1
Π
v
,
w
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0],
we
obtain
an
outomorphism
β
v
,
w
of
Π
2
|
Π
v
,
w
which,
by
construction,
satisfies
the
four
conditions
listed
in
assertion
(v).
This
completes
the
proof
of
assertion
(v).
Lemma
4.13
(Glueability
of
combinatorial
cuspidalizations
in
the
case
of
precisely
one
node).
Suppose
that
n
=
2,
and
that
[cf.
Definition
4.11]
is
surjective.
#Node(G)
=
1.
Then
ρ
brch
2
Proof.
If
G
is
noncyclically
primitive
[cf.
[CbTpI],
Definition
4.1],
then
the
surjectivity
of
ρ
brch
follows
immediately
from
Lemma
4.12,
(v)
[cf.
2
also
[CmbCsp],
Proposition
1.2,
(i)],
together
with
the
fact
that
the
natural
injection
Π
v
,
w
→
Π
1
is
an
isomorphism
[cf.
Lemma
4.12,
(ii)].
in
the
case
where
Thus,
it
remains
to
verify
the
surjectivity
of
ρ
brch
2
G
is
cyclically
primitive
[cf.
[CbTpI],
Definition
4.1].
Since
we
are
in
the
situation
of
[CbTpI],
Lemma
4.3,
we
shall
apply
the
notational
conventions
established
in
[CbTpI],
Lemma
4.3.
Also,
we
shall
write
COMBINATORIAL
ANABELIAN
TOPICS
II
125
Vert(G)
=
{v},
Node(G)
=
{e}.
Let
x
∈
X
2
(k)
be
a
k-rational
geomet-
ric
point
of
X
2
such
that
x
{1}
∈
X(k)
[cf.
Definition
3.1,
(i)]
lies
on
the
unique
node
of
X
log
[i.e.,
which
corresponds
to
e].
Recall
from
[CbTpI],
Lemma
4.3,
(i),
that
we
have
a
natural
exact
sequence
1
−→
π
1
temp
(G
∞
)
−→
π
1
temp
(G)
−→
π
1
top
(G)
−→
1
.
∞
∈
π
1
temp
(G)
Let
γ
∞
∈
π
1
top
(G)
be
a
generator
of
π
1
top
(G)
(≃
Z)
and
γ
∼
a
lifting
of
γ
∞
.
By
abuse
of
notation,
write
γ
∞
∈
Π
G
←
Π
1
for
the
∼
image
of
γ
∞
∈
π
1
temp
(G)
via
the
natural
injection
π
1
temp
(G)
→
Π
G
←
Π
1
[cf.
the
evident
pro-Σ
generalization
of
[SemiAn],
Proposition
3.6,
(iii);
[RZ],
Proposition
3.3.15].
Next,
let
us
fix
a
verticial
subgroup
temp
(G
∞
)
⊆)
π
1
temp
(G)
Π
temp
v
(0)
⊆
(π
1
that
lifts
the
of
π
1
temp
(G)
that
corresponds
to
a
vertex
v
(0)
∈
Vert(
G)
vertex
V
(0)
∈
Vert(G
∞
)
[cf.
[CbTpI],
Lemma
4.3,
(iii)].
Thus,
for
each
a
∞
,
we
obtain
a
integer
a
∈
Z,
by
forming
the
conjugate
of
Π
temp
v
(0)
by
γ
verticial
subgroup
temp
(G
∞
)
⊆)
π
1
temp
(G)
Π
temp
v
(a)
⊆
(π
1
that
lifts
the
of
π
1
temp
(G)
associated
to
some
vertex
v
(a)
∈
Vert(
G)
vertex
V
(a)
∈
Vert(G
∞
)
[cf.
[CbTpI],
Lemma
4.3,
(iii),
(vi)].
Write
Π
v
(a)
⊆
Π
G
temp
for
the
image
of
Π
temp
(G)
in
Π
G
.
v
(a)
⊆
π
1
Next,
let
us
suppose
that
γ
∞
was
chosen
in
such
a
way
that,
for
each
a
∈
Z,
the
intersection
N
(
v
(a))
∩
N
(
v
(a
+
1))
consists
of
a
unique
node
n
(a,
a
+
1)
∈
Node(
G)
that
lifts
the
node
N
(a
+
1)
∈
Node(G
∞
)
[cf.
[CbTpI],
Lemma
4.3,
(iii)].
[One
verifies
easily
that
such
a
γ
∞
always
exists.]
Then
let
us
observe
that,
for
each
a
≤
b
∈
Z,
we
have
a
nat-
ural
morphism
of
semi-graphs
of
anabelioids
G
[a,b]
→
G
∞
[cf.
[CbTpI],
Lemma
4.3,
(iv)],
which
induces
injections
[cf.
the
evident
pro-Σ
gen-
eralizations
of
[SemiAn],
Example
2.10;
[SemiAn],
Proposition
2.5,
(i);
[SemiAn],
Proposition
3.6,
(iii);
[RZ],
Proposition
3.3.15]
π
1
temp
(G
[a,b]
)
→
π
1
temp
(G
∞
),
Π
G
[a,b]
→
Π
G
—
where
we
write,
respectively,
π
1
temp
(G
[a,b]
),
Π
G
[a,b]
for
the
tempered,
pro-Σ
fundamental
groups
of
the
semi-graph
of
anabelioids
G
[a,b]
of
pro-Σ
PSC-type
—
which
are
well-defined
up
to
composition
with
in-
ner
automorphisms.
By
choosing
appropriate
basepoints
[cf.
also
our
choice
of
γ
∞
],
these
inner
automorphism
indeterminacies
may
be
elimi-
nated
in
such
a
way
that,
for
each
a
≤
c
≤
b,
the
resulting
injections
are
compatible
with
one
another
and,
moreover,
their
images
contain
the
∼
temp
(G
∞
),
Π
v
(c)
⊆
Π
G
←
Π
1
,
respectively.
Then,
subgroups
Π
temp
v
(c)
⊆
π
1
126
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
relative
to
the
resulting
inclusions,
Π
temp
(c)
form
verticial
subgroups
v
(c)
,
Π
v
temp
of
π
1
(G
[a,b]
),
Π
G
[a,b]
associated
to
the
vertex
of
G
[a,b]
corresponding
to
V
(c)
[cf.
[CbTpI],
Lemma
4.3,
(iii)].
In
particular,
we
have
a
natural
isomorphism
∼
def
Π
[a,a+1]
=
Π
v
(a),
v
(a+1)
−→
Π
G
[a,a+1]
[cf.
Lemma
4.12,
(ii)].
Let
us
write
def
Π
2
|
[a,a+1]
=
Π
2
|
Π
[a,a+1]
⊆
Π
2
[cf.
Lemma
4.12,
(ii)];
def
Π
[a]
=
Π
v
(a)
;
def
Π
2
|
[a]
=
Π
2
×
Π
1
Π
[a]
⊆
Π
2
|
[a−1,a]
,
Π
2
|
[a,a+1]
.
Next,
we
claim
that
the
following
assertion
holds:
Claim
4.13.A:
The
profinite
group
Π
G
is
topologically
generated
by
Π
[0]
⊆
Π
G
and
γ
∞
∈
Π
G
.
Indeed,
let
us
first
observe
that
it
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
[CmbCsp],
Propo-
sition
1.5,
(iii)
[i.e.,
in
essence,
from
the
“van
Kampen
Theorem”
in
elementary
algebraic
topology],
that
•
the
image
of
the
natural
homomorphism
lim
π
1
temp
(G
[−a,a]
)
−→
π
1
temp
(G
∞
)
−→
a≥0
—
where
the
inductive
limit
is
taken
in
the
category
of
tem-
pered
groups
[cf.
[SemiAn],
Definition
3.1,
(i);
[SemiAn],
Ex-
ample
2.10;
[SemiAn],
Proposition
3.6,
(i)]
—
is
dense;
•
for
each
nonnegative
integer
a,
the
tempered
group
π
1
temp
(G
[−a,a]
)
[cf.
[SemiAn],
Example
2.10;
[SemiAn],
Proposition
3.6,
(i)]
is
temp
temp
(G
[−a,a]
).
topologically
generated
by
Π
temp
v
(−a)
,
.
.
.
,
Π
v
(a)
⊆
π
1
In
particular,
it
follows
immediately
from
the
exact
sequence
of
[CbTpI],
Lemma
4.3,
(i),
that
the
tempered
group
π
1
temp
(G)
[cf.
[SemiAn],
Ex-
ample
2.10;
[SemiAn],
Proposition
3.6,
(i)]
is
topologically
generated
temp
(G)
and
γ
∞
∈
π
1
temp
(G).
Thus,
Claim
4.13.A
fol-
by
Π
temp
v
(0)
⊆
π
1
lows
immediately
from
the
fact
that
the
image
of
the
natural
injection
π
1
temp
(G)
→
Π
G
is
dense.
This
completes
the
proof
of
Claim
4.13.A.
For
a
∈
Z,
let
us
write
[a,a+1]
def
G
2/1
=
G
2∈{1,2},x
[cf.
Definition
3.1,
(iii)],
where
we
fix
isomorphisms
∼
Π
2/1
−→
Π
G
[a,a+1]
,
2/1
∼
Π
{2}
−→
Π
G
2∈{2},x
=
Π
G
[the
latter
of
which
is
to
be
understood
as
being
independent
of
a
∈
Z]
as
in
[i.e.,
that
belong
to
the
collections
of
isomorphisms
that
constitute
COMBINATORIAL
ANABELIAN
TOPICS
II
127
the
outer
isomorphisms
of
the
final
display
of]
Definition
3.1,
(iii),
to
be
isomorphisms
[cf.
the
discussion
of
the
final
portion
of
Lemma
4.12,
[a,a+1]
(v)]
such
that
the
semi-graph
of
anabelioids
structure
on
G
2/1
is
the
semi-graph
of
anabelioids
structure
determined
by
the
resulting
composite
∼
∼
Π
n
(a,a+1)
→
Π
G
←
Π
1
→
Out(Π
2/1
)
→
Out(Π
G
[a,a+1]
)
2/1
—
where
we
write
Π
n
(a,a+1)
⊆
Π
G
for
the
nodal
subgroup
of
Π
G
as-
sociated
to
the
unique
element
n
(a,
a
+
1)
∈
N
(
v
(a))
∩
N
(
v
(a
+
1)),
and
the
third
arrow
arises
from
the
outer
action
determined
by
the
p
Π
2/1
exact
sequence
1
→
Π
2/1
→
Π
2
→
Π
1
→
1
—
in
a
fashion
compatible
with
the
projection
p
Π
{1,2}/{2}
|
Π
2/1
:
Π
2/1
Π
{2}
and
the
isomorphisms
∼
∼
Π
{2}
→
Π
G
←
Π
1
[cf.
Definition
3.1,
(ii)].
Here,
we
note
that,
for
a,
∼
[a,a+1]
∼
[b,b+1]
→
G
2∈{1,2},x
→
G
2/1
b
∈
Z,
there
exist
natural
isomorphisms
G
2/1
of
semi-graphs
of
anabelioids
of
pro-Σ
PSC-type
[induced
by
conjuga-
b−a
tion
by
γ
∞
].
On
the
other
hand,
it
is
not
difficult
to
show
[although
we
shall
not
use
this
fact
in
the
present
proof!]
that
the
well-known
injectivity
of
the
homomorphism
Π
1
→
Out(Π
2/1
)
of
the
above
display
[cf.
[CbTpI],
Lemma
5.4,
(i),
(ii),
(iii);
[CbTpI],
Theorem
4.8,
(iv);
[Asd],
Theorem
1;
[Asd],
the
Remark
following
the
proof
of
Theorem
1;
[CmbGC],
Proposition
1.2,
(i),
(ii)]
implies
that
when
a
=
b,
the
composite
∼
∼
Π
G
[a,a+1]
←
Π
2/1
→
Π
G
[b,b+1]
2/1
2/1
in
fact
fails
to
be
graphic!
For
each
a
∈
Z,
let
us
write
[a,a+1][a]
def
G
2/1
[a,a+1]
=
(G
2/1
[a,a+1][a+1]
def
)
{e
v(a)
◦
}
,
G
2/1
[a,a+1]
=
(G
2/1
)
{e
v(a+1)
◦
}
—
where
we
write
e
v(a)
◦
,
e
v(a+1)
◦
for
the
nodes
“e
z
◦
”
of
Lemma
4.12,
(iii),
that
occur,
respectively,
in
the
cases
where
the
pair
“(G
2/1
,
z
◦
)”
[a,a+1]
[a,a+1]
is
taken
to
be
(G
2/1
,
v
(a)
◦
);
(G
2/1
,
v
(a
+
1)
◦
).
Then
one
verifies
easily
[cf.
Lemma
4.12,
(i),
(iii)]
that
the
composite
∼
∼
∼
∼
Π
G
[a−1,a][a]
→
Π
G
[a−1,a]
←
Π
2/1
→
Π
G
[a,a+1]
←
Π
G
[a,a+1][a]
2/1
2/1
2/1
2/1
—
where
the
first
and
fourth
arrows
are
the
natural
specialization
outer
isomorphisms
[cf.
[CbTpI],
Definition
2.10],
and
the
second
and
third
arrows
are
the
isomorphisms
fixed
above
—
is
graphic.
In
light
of
this
observation,
it
makes
sense
to
write
[a]
def
[a−1,a][a]
∼
G
2/1
=
G
2/1
[a,a+1][a]
→
G
2/1
[cf.
Figure
4
below].
This
notation
allows
us
to
express
the
graphicity
observed
above
in
the
following
way:
128
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
The
composites
∼
∼
∼
Π
[a]
→
Π
G
←
Π
1
→
Out(Π
2/1
)
→
Out(Π
G
[a−1,a]
)
←
Out(Π
G
[a]
),
2/1
2/1
∼
∼
∼
Π
[a]
→
Π
G
←
Π
1
→
Out(Π
2/1
)
→
Out(Π
G
[a,a+1]
)
←
Out(Π
G
[a]
)
2/1
2/1
—
where
the
third
arrows
in
each
line
of
the
display
arise
from
the
outer
action
determined
by
the
exact
p
Π
2/1
sequence
1
→
Π
2/1
→
Π
2
→
Π
1
→
1,
the
fourth
arrows
are
the
isomorphisms
induced
by
the
isomor-
∼
∼
phisms
Π
2/1
→
Π
G
[a−1,a]
and
Π
2/1
→
Π
G
[a,a+1]
fixed
2/1
2/1
above,
and
the
fifth
arrows
are
the
isomorphisms
in-
duced
by
the
natural
specialization
outer
isomorphisms
[cf.
[CbTpI],
Definition
2.10]
—
factor
through
[a]
Aut(G
2/1
)
⊆
Out(Π
G
[a]
).
2/1
[a−1,a]
[a]
G
2/1
•
•
··
··
··
··
··
·
G
2/1
•
•
··
··
··
··
··
·
∨
∨
v
(a)
v
(a
+
1)
•
Π
[a−1,a]
[a−1,a]
Figure
4:
G
2/1
•
··
··
··
··
··
·
∨
v
(a
−
1)
···
•
[a,a+1]
G
2/1
Π
[a,a+1]
[a]
•
···
[a,a+1]
,
G
2/1
,
and
G
2/1
Now
we
turn
to
the
verification
of
the
surjectivity
of
the
homomor-
phism
ρ
brch
.
Let
α
v
∈
Glu(Π
2
)
(⊆
Out
FC
((Π
v
)
2
)
G-node
).
Write
(α
v
)
1
∈
2
Glu(Π
1
)
for
the
image
of
α
v
∈
Glu(Π
2
)
via
the
injection
of
Lemma
4.10,
(i).
Let
α
1
∈
Aut
|Brch(G)|
(G)
be
such
that
ρ
brch
(α
1
)
=
(α
v
)
1
∈
Glu(Π
1
)
1
COMBINATORIAL
ANABELIAN
TOPICS
II
129
[cf.
Theorem
4.2,
(iii);
Definition
4.11].
Now,
by
applying
Lemma
4.12,
(v),
in
the
case
where
we
take
the
pair
“(
v
,
w)”
to
be
(
v
(0),
v
(1)),
we
def
obtain
an
outomorphism
β
[0,1]
=
β
v
(0),
v
(1)
[α
1
]
[cf.
Lemma
4.12,
(v)]
of
Π
2
|
[0,1]
[cf.
the
notation
of
the
discussion
preceding
Claim
4.13.A].
Let
†
•
β
[0,1]
∈
Aut(Π
2
|
[0,1]
)
be
an
automorphism
that
lifts
β
[0,1]
∈
Out(Π
2
|
[0,1]
)
and
preserves
the
subgroup
Π
n
(0,1)
⊆
Π
[0,1]
[cf.
condition
(4)
of
Lemma
4.12,
(v)]
and
∞
∈
Π
1
.
•
γ
∞
∈
Π
2
a
lifting
of
γ
Then
since
[as
is
easily
verified]
Π
2
|
[1,2]
[cf.
the
notation
of
the
dis-
cussion
preceding
Claim
4.13.A]
is
the
conjugate
of
Π
2
|
[0,1]
by
γ
∞
,
by
†
conjugating
β
[0,1]
by
the
inner
automorphism
determined
by
γ
∞
,
we
obtain
an
automorphism
β
†
of
Π
2
|
[1,2]
,
whose
associated
outomor-
[1,2]
phism
we
denote
by
β
[1,2]
.
Now
we
claim
that
the
following
assertion
holds:
Claim
4.13.B:
There
exist
automorphisms
β
[0,1]
,
β
[1,2]
of
Π
2
|
[0,1]
,
Π
2
|
[1,2]
that
lift
β
[0,1]
,
β
[1,2]
,
respectively,
such
that
(i)
the
outomorphisms
of
Π
2/1
(⊆
Π
2
|
[0,1]
,
Π
2
|
[1,2]
)
de-
termined
by
β
[0,1]
,
β
[1,2]
coincide;
(ii)
the
automorphism
of
Π
[0,1]
determined
by
the
au-
tomorphism
β
[0,1]
preserves
the
subgroups
Π
n
(0,1)
,
Π
[0]
,
Π
[1]
⊆
Π
[0,1]
;
†
(iii)
β
[0,1]
=
β
[0,1]
,
and
β
[1,2]
is
the
post-composite
of
β
†
with
an
inner
automorphism
arising
from
an
[1,2]
element
of
Π
2
|
[1]
.
†
Indeed,
observe
that
there
exist
automorphisms
β
[0,1]
,
β
[1,2]
[e.g.,
β
[0,1]
,
†
β
]
of
Π
2
|
[0,1]
,
Π
2
|
[1,2]
that
lift
β
[0,1]
,
β
[1,2]
,
respectively,
such
that
[1,2]
•
the
outomorphisms
(
β
[0,1]
)
2/1
,
(
β
[1,2]
)
2/1
of
Π
2/1
determined
by
β
[0,1]
,
β
[1,2]
are
contained
in
[0,1]
Aut
|Brch(G
2/1
)|
(G
2/1
),
[0,1]
[1,2]
Aut
|Brch(G
2/1
)|
(G
2/1
)
(⊆
Out(Π
2/1
)),
[1,2]
respectively,
and,
•
conditions
(ii),
(iii)
of
Claim
4.13.B
are
satisfied
[cf.
the
discussion
of
the
final
portion
of
Lemma
4.12,
(v);
Lemma
4.12,
(v),
(1);
[CmbGC],
Proposition
1.5,
(i)].
In
particular,
it
follows
that,
∼
relative
to
the
specialization
outer
isomorphisms
Π
G
[1]
→
Π
G
[0,1]
,
Π
G
[1]
2/1
∼
2/1
2/1
→
Π
G
[1,2]
that
appeared
in
the
discussion
following
the
proof
of
Claim
2/1
4.13.A,
together
with
the
natural
inclusion
of
[CbTpI],
Proposition
2.9,
130
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(ii),
[1]
(
β
[0,1]
)
2/1
,
(
β
[1,2]
)
2/1
∈
Aut
|Brch(G
2/1
)|
(G
2/1
)
(⊆
Out(Π
2/1
))
.
[1]
Moreover,
it
follows
immediately
from
condition
(3)
of
Lemma
4.12,
(v),
applied
in
the
case
of
β
[0,1]
,
together
with
the
definition
of
β
[1,2]
,
that
the
outomorphisms
of
the
configuration
space
subgroup
Π
2
⊇
Π
2
|
[0,1]
⊇
⊆
Π
2
|
[1,2]
⊆
Π
2
(Π
v
(1)
)
2
associated
to
the
vertex
v
(1)
determined
by
β
[0,1]
,
β
[1,2]
coincide
with
α
v
.
Now
let
us
recall
from
the
above
discussion
that
the
composite
∼
Π
[1]
→
Π
1
→
Out(Π
2/1
)
→
Out(Π
G
[1]
)
2/1
factors
through
[1]
Aut(G
2/1
)
⊆
Out(Π
G
[1]
)
.
2/1
Thus,
it
follows
immediately
from
the
displayed
exact
sequence
of
The-
orem
4.2,
(iii)
[cf.
also
Remark
4.9.1],
that
—
after
possibly
replacing
β
[1,2]
by
the
post-composite
of
β
[1,2]
with
an
inner
automorphism
arising
from
a
suitable
element
of
Π
2
|
[1]
[which
does
not
affect
the
validity
of
conditions
(ii),
(iii)
of
Claim
4.13.B]
—
if
we
write
[1]
def
[1]
|Brch(G
2/1
)|
δ
=
(
β
[0,1]
)
2/1
◦
(
β
[1,2]
)
−1
(G
2/1
)
(⊆
Out(Π
2/1
))
,
2/1
∈
Aut
[1]
then
it
holds
that
δ
∈
Dehn(G
2/1
).
Next,
let
us
observe
that,
for
a
∈
{0,
1},
since
β
[a,a+1]
preserves
the
Π
2/1
-conjugacy
class
of
cuspidal
inertia
subgroups
associated
to
the
di-
agonal
cusp
[cf.
condition
(3)
of
Lemma
4.12,
(v)],
it
follows
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
[CmbCsp],
Proposition
1.2,
(iii),
that
the
outomorphism
(
β
[a,a+1]
)
{2}
of
Π
{2}
in-
duced
by
β
[a,a+1]
on
the
quotient
∼
Π
G
[1]
←
Π
2/1
→
Π
2
2/1
p
Π
{1,2}/{2}
Π
{2}
∼
is
compatible,
relative
to
the
natural
inclusion
Π
[a,a+1]
→
Π
1
→
Π
{2}
,
with
the
outomorphism
α
1
|
Π
[a,a+1]
[cf.
condition
(4)
of
Lemma
4.12,
(v)].
Since
an
element
of
Aut
|Brch(G)|
(G)
is
completely
determined
by
its
restriction
to
Aut(G
[a,a+1]
)
[cf.
[CbTpI],
Definition
4.4;
[CbTpI],
Remark
4.8.1],
we
thus
conclude
that,
relative
to
the
natural
outer
∼
∼
isomorphisms
Π
{2}
→
Π
1
→
Π
G
,
it
holds
that
(
β
[a,a+1]
)
{2}
=
α
1
.
In
particular,
it
follows
that
the
element
of
Aut
|Brch(G)|
(G)
induced
by
δ
∈
Aut
[1]
|Brch(G
2/1
)|
[1]
(G
2/1
)
on
the
quotient
Π
G
[1]
2/1
∼
←
Π
2/1
→
Π
2
p
Π
{1,2}/{2}
COMBINATORIAL
ANABELIAN
TOPICS
II
131
∼
Π
{2}
→
Π
G
is
trivial.
On
the
other
hand,
let
us
observe
that
one
verifies
easily
from
[CbTpI],
Theorem
4.8,
(iii),
(iv),
that
this
composite
∼
Π
G
[1]
←
Π
2/1
→
Π
2
p
Π
{1,2}/{2}
2/1
∼
Π
{2}
→
Π
G
determines
an
isomorphism
[1]
∼
Dehn(G
2/1
)
−→
Dehn(G)
.
Thus,
we
conclude
that
δ
is
the
identity
outomorphism
of
Π
2/1
.
In
particular,
condition
(i)
of
Claim
4.13.B
is
satisfied.
This
completes
the
proof
of
Claim
4.13.B.
Next,
let
us
fix
an
automorphism
α
1
∈
Aut(Π
1
)
that
lifts
α
1
∈
∼
Aut
|Brch(G)|
(G)
⊆
Out(Π
G
)
←
Out(Π
1
)
and
preserves
the
subgroup
Π
n
(0,1)
⊆
Π
1
[hence
also
the
subgroups
Π
[0]
,
Π
[1]
,
Π
[0,1]
⊆
Π
1
],
and
whose
restriction
to
Π
[0,1]
⊆
Π
1
coincides
with
the
automorphism
of
Π
[0,1]
de-
termined
by
the
automorphism
β
[0,1]
of
Π
2
|
[0,1]
.
[One
verifies
easily
that
such
an
α
1
always
exists
—
cf.
Lemma
4.12,
(v),
(4);
Claim
4.13.B,
(ii).]
Write
β
2/1
∈
Out(Π
2/1
)
for
the
outomorphism
of
Π
2/1
⊆
Π
2
|
[0,1]
determined
by
β
[0,1]
[or,
equivalently,
β
[1,2]
—
cf.
Claim
4.13.B,
(i)].
Now
we
claim
that
the
following
assertion
holds:
Claim
4.13.C:
Write
ρ
:
Π
1
→
Out(Π
2/1
)
for
the
ho-
momorphism
determined
by
the
exact
sequence
1
→
p
Π
2/1
Π
2/1
→
Π
2
→
Π
1
→
1.
Then
−1
ρ(
α
1
(
γ
∞
))
=
β
2/1
◦
ρ(
γ
∞
)
◦
β
2/1
∈
Out(Π
2/1
)
.
Indeed,
let
us
first
observe
that
it
follows
from
conditions
(i)
and
(iii)
†
of
Claim
4.13.B,
together
with
the
definition
of
β
[1,2]
,
that
there
exists
an
element
∈
Π
[1]
such
that
−1
−1
γ
∞
)
◦
β
2/1
=
ρ(
−1
)
ρ(
γ
∞
)
◦
β
2/1
◦
ρ(
(∗
1
)
.
Next,
let
us
observe
that
if
we
write
−1
η
=
α
1
(
γ
∞
)
·
γ
∞
∈
Π
1
def
(∗
2
),
then
it
follows
immediately
from
the
commensurable
terminality
of
Π
[1]
in
Π
1
[cf.
[CmbGC],
Proposition
1.2,
(ii)],
together
with
our
choices
of
∞
—
which
imply
that
α
1
and
γ
−1
α
1
(
γ
∞
)
·
γ
∞
·
Π
[1]
·
γ
∞
·
α
1
(
γ
∞
)
−1
=
=
=
=
=
α
1
(
γ
∞
)
·
Π
[0]
·
α
1
(
γ
∞
)
−1
α
1
(
γ
∞
)
·
α
1
(Π
[0]
)
·
α
1
(
γ
∞
)
−1
−1
α
1
(
γ
∞
·
Π
[0]
·
γ
∞
)
α
1
(Π
[1]
)
Π
[1]
—
that
η
∈
Π
[1]
.
Thus,
to
verify
Claim
4.13.C,
it
suffices
to
verify
that
ρ(
)
=
ρ(η).
132
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
To
this
end,
let
ζ
∈
Π
[0]
.
Then,
by
our
choice
of
γ
∞
,
it
follows
that
−1
∞
∈
Π
[1]
.
In
particular,
since
the
outomorphism
β
2/1
arises
γ
∞
·
ζ
·
γ
from
an
automorphism
β
[0,1]
of
Π
2
|
[0,1]
,
which
is
an
automorphism
over
the
restriction
of
α
1
to
Π
[0,1]
,
it
follows
immediately
that
α
1
(ζ))
◦
β
2/1
β
2/1
◦
ρ(ζ)
=
ρ(
(∗
3
)
.
−1
−1
β
2/1
◦
ρ(
γ
∞
·
ζ
·
γ
∞
)
=
ρ(
α
1
(
γ
∞
·
ζ
·
γ
∞
))
◦
β
2/1
(∗
4
)
.
Thus,
if
we
write
−1
Θ
=
ρ(
·
γ
∞
·
α
1
(ζ)
·
γ
∞
·
−1
)
◦
β
2/1
∈
Out(Π
2/1
),
def
−1
∞
·
α
1
(ζ)
·
γ
∞
·
η
−1
)
◦
β
2/1
∈
Out(Π
2/1
),
Θ
η
=
ρ(η
·
γ
def
then
Θ
=
=
=
=
=
−1
ρ(
·
γ
∞
·
α
1
(ζ))
◦
β
2/1
◦
ρ(
γ
∞
)
−1
∞
)
ρ(
·
γ
∞
)
◦
β
2/1
◦
ρ(ζ
·
γ
−1
β
2/1
◦
ρ(
γ
∞
·
ζ
·
γ
∞
)
−1
γ
∞
·
ζ
·
γ
∞
))
◦
β
2/1
ρ(
α
1
(
Θ
η
[cf.
(∗
1
)]
[cf.
(∗
3
)]
[cf.
(∗
1
)]
[cf.
(∗
4
)]
[cf.
(∗
2
)]
−1
—
which
thus
implies
that
ρ(η
−1
·
)
commutes
with
ρ(
γ
∞
·
α
1
(ζ)·
γ
∞
).
In
−1
−1
particular,
since
γ
∞
·
α
1
(Π
[0]
)·
γ
∞
=
γ
∞
·Π
[0]
·
γ
∞
=
Π
[1]
,
by
allowing
“ζ”
to
vary
among
the
elements
of
Π
[0]
,
it
follows
that
ρ(η
−1
·
)
centralizes
ρ(Π
[1]
).
On
the
other
hand,
it
follows
from
[Asd],
Theorem
1;
[Asd],
the
Remark
following
the
proof
of
Theorem
1,
that
ρ
is
injective.
Thus,
since
,
η
∈
Π
[1]
,
we
conclude
that
η
−1
·
∈
Z(Π
[1]
)
=
{1}
[cf.
[CmbGC],
Remark
1.1.3].
This
completes
the
proof
of
Claim
4.13.C.
Now
let
us
recall
that
the
outomorphism
β
2/1
of
Π
2/1
of
Claim
4.13.C
arises
from
an
automorphism
β
[0,1]
of
Π
2
|
[0,1]
.
Thus,
it
follows
immedi-
ately
from
Claims
4.13.A,
4.13.C
that
the
outomorphism
β
2/1
of
Π
2/1
is
compatible
with
the
automorphism
α
1
∈
Aut(Π
1
)
relative
to
the
homomorphism
Π
1
→
Out(Π
2/1
)
determined
by
the
exact
sequence
p
Π
2/1
1
→
Π
2/1
→
Π
2
→
Π
1
→
1.
In
particular
—
by
considering
the
∼
out
natural
isomorphism
Π
2
→
Π
2/1
Π
1
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
—
we
conclude
that
the
outo-
morphism
β
2/1
∈
Out(Π
2/1
)
extends
to
an
outomorphism
α
2
of
Π
2
.
On
the
other
hand,
it
follows
immediately
from
the
various
defini-
(α
2
)
=
α
v
∈
Glu(Π
2
)
[cf.
condition
(3)
of
tions
involved
that
ρ
brch
2
Lemma
4.12,
(v)],
and
that
α
2
∈
Out
FC
(Π
2
)
brch
[cf.
condition
(2)
of
Lemma
4.12,
(v);
[CmbCsp],
Proposition
1.2,
(i)].
This
completes
the
proof
of
Lemma
4.13
in
the
case
where
G
is
cyclically
primitive,
hence
also
of
Lemma
4.13.
COMBINATORIAL
ANABELIAN
TOPICS
II
133
Theorem
4.14
(Glueability
of
combinatorial
cuspidalizations).
Let
(g,
r)
be
a
pair
of
nonnegative
integers
such
that
2g
−
2
+
r
>
0;
n
a
positive
integer;
Σ
a
set
of
prime
numbers
which
is
either
equal
to
the
set
of
all
prime
numbers
or
of
cardinality
one;
k
an
algebraically
closed
field
of
characteristic
∈
Σ;
(Spec
k)
log
the
log
scheme
obtained
by
equipping
Spec
k
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
=
X
1
log
a
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
.
Write
G
for
the
semi-graph
of
anabelioids
of
pro-
Σ
PSC-type
determined
by
the
stable
log
curve
X
log
.
For
each
positive
integer
i,
write
X
i
log
for
the
i-th
log
configuration
space
of
the
stable
log
curve
X
log
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”];
Π
i
for
the
maximal
pro-Σ
quotient
of
the
kernel
of
the
natural
surjection
π
1
(X
i
log
)
π
1
((Spec
k)
log
).
Then
the
following
hold:
(i)
There
exists
a
natural
commutative
diagram
of
profinite
groups
ρ
brch
n+1
Out
FC
(Π
n+1
)
brch
−−−→
Glu(Π
n+1
)
⏐
⏐
⏐
⏐
ρ
brch
Out
FC
(Π
n
)
brch
−−
n
−→
Glu(Π
n
)
[cf.
Definition
4.6,
(i);
Definition
4.9;
Lemma
4.10,
(i);
Defi-
nition
4.11]
—
where
the
vertical
arrows
are
injective.
(ii)
The
closed
subgroup
Dehn(G)
⊆
(Aut(G)
⊆)
Out(Π
1
)
[cf.
[CbTpI],
Definition
4.4]
is
contained
in
the
image
of
the
injection
Out
FC
(Π
n
)
brch
→
Out
FC
(Π
1
)
brch
[cf.
the
left-hand
vertical
ar-
rows
of
the
diagrams
of
(i),
for
varying
n].
Thus,
one
may
regard
Dehn(G)
as
a
closed
subgroup
of
Out
FC
(Π
n
)
brch
,
i.e.,
Dehn(G)
⊆
Out
FC
(Π
n
)
brch
.
(iii)
The
homomorphism
ρ
brch
:
Out
FC
(Π
n
)
brch
→
Glu(Π
n
)
of
(i)
n
and
the
inclusion
Dehn(G)
→
Out
FC
(Π
n
)
brch
of
(ii)
fit
into
an
exact
sequence
of
profinite
groups
ρ
brch
n
1
−→
Dehn(G)
−→
Out
FC
(Π
n
)
brch
−→
Glu(Π
n
)
−→
1
.
In
particular,
the
commutative
diagram
of
(i)
is
cartesian,
and
the
horizontal
arrows
of
this
diagram
are
surjective.
Proof.
Assertion
(i)
follows
immediately
from
Lemma
4.10,
(i),
together
with
the
injectivity
portion
of
[NodNon],
Theorem
B.
Assertion
(ii)
follows
immediately
from
Proposition
3.24,
(ii);
Theorem
4.2,
(i).
Finally,
we
verify
assertion
(iii).
First,
we
claim
that
the
following
assertion
holds:
Claim
4.14.A:
Ker(ρ
brch
n
)
=
Dehn(G)
[cf.
assertion
(ii)].
134
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Indeed,
it
follows
immediately
from
Theorem
4.2,
(iii)
[cf.
also
Re-
mark
4.9.1],
together
with
assertion
(i),
that
we
have
a
natural
com-
mutative
diagram
ρ
brch
FC
brch
−−
n
−→
Glu(Π
n
)
1
−−−→
Ker(ρ
brch
n
)
−−−→
Out
(Π
n
)
⏐
⏐
⏐
⏐
⏐
⏐
ρ
brch
1
−−−→
Dehn(G)
−−−→
Out
FC
(Π
1
)
brch
−−
1
−→
Glu(Π
1
)
−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
injective.
Thus,
Claim
4.14.A
follows
immediately.
In
particular,
to
complete
the
verification
of
assertion
(iii),
it
suffices
to
verify
the
surjectivity
of
ρ
brch
n
.
The
remainder
of
the
proof
of
assertion
(iii)
is
devoted
to
verifying
this
surjectivity.
Next,
we
claim
that
the
following
assertion
holds:
Claim
4.14.B:
If
n
=
2,
then
ρ
brch
is
surjective.
n
We
verify
Claim
4.14.B
by
induction
on
#Node(G).
If
#Node(G)
=
0,
then
Claim
4.14.B
is
immediate.
If
#Node(G)
=
1,
then
Claim
4.14.B
follows
from
Lemma
4.13.
Now
suppose
that
#Node(G)
>
1,
and
that
the
induction
hypothesis
is
in
force.
Let
(α
v
)
v∈Vert(G)
∈
Glu(Π
2
).
Write
((α
v
)
1
)
v∈Vert(G)
∈
Glu(Π
1
)
for
the
element
of
Glu(Π
1
)
determined
by
(α
v
)
v∈Vert(G)
[i.e.,
the
image
of
(α
v
)
v∈Vert(G)
via
the
right-hand
vertical
arrow
of
the
diagram
of
assertion
(i)
in
the
case
where
n
=
1].
Let
e
∈
Node(G).
Write
H
for
the
unique
sub-semi-graph
of
PSC-type
[cf.
[CbTpI],
Definition
2.2,
(i)]
of
the
underlying
semi-graph
of
G
whose
set
def
of
vertices
is
V(e).
Then
one
verifies
easily
that
S
=
Node(G|
H
)
\
{e}
[cf.
[CbTpI],
Definition
2.2,
(ii)]
is
not
of
separating
type
[cf.
[CbTpI],
Definition
2.5,
(i)]
as
a
subset
of
Node(G|
H
).
Thus,
since
(G|
H
)
S
[cf.
[CbTpI],
Definition
2.5,
(ii)]
has
precisely
one
node,
and
(α
v
)
v∈V(e)
may
be
regarded
as
an
element
of
Glu((Π
H,S
)
2
)
—
where
we
use
the
notation
(Π
H,S
)
2
to
denote
a
configuration
space
subgroup
of
Π
2
associated
to
(H,
S)
[cf.
Definition
4.3],
to
which
the
notation
“Glu(−)”
is
applied
in
the
evident
sense
—
it
follows
from
Lemma
4.13
that
there
exists
an
outomorphism
β
H,S
of
(Π
H,S
)
2
⊆
Π
2
that
lifts
(α
v
)
v∈V(e)
∈
Glu((Π
H,S
)
2
).
Next,
let
us
observe
that
it
follows
immediately
from
the
various
definitions
involved
that
def
γ
=
(β
H,S
,
(α
v
)
v
∈V(e)
)
∈
Out((Π
H,S
)
2
)
×
Out((Π
v
)
2
)
v
∈V(e)
may
be
regarded
as
an
element
of
the
“Glu(Π
2
)”
that
occurs
in
the
case
where
we
take
the
stable
log
curve
“X
log
”
to
be
a
stable
log
curve
over
(Spec
k)
log
obtained
by
deforming
the
node
corresponding
to
e.
Thus,
since
the
number
of
nodes
of
such
a
stable
log
curve
is
=
#Node(G)
−
1
<
#Node(G),
by
applying
the
induction
hypothesis,
we
conclude
that
the
above
γ
arises
from
an
outomorphism
α
γ
∈
Out
FC
(Π
2
)
brch
.
COMBINATORIAL
ANABELIAN
TOPICS
II
135
On
the
other
hand,
it
follows
immediately
from
the
various
definitions
coincides
with
(α
v
)
v∈Vert(G)
.
This
involved
that
the
image
of
α
γ
via
ρ
brch
2
completes
the
proof
of
Claim
4.14.B.
Finally,
we
verify
the
surjectivity
of
ρ
brch
[for
arbitrary
n]
by
in-
n
duction
on
n.
If
n
≤
2,
then
the
surjectivity
of
ρ
brch
follows
from
n
Theorem
4.2,
(iii)
[cf.
also
Remark
4.9.1],
Claim
4.14.B.
Now
sup-
pose
that
n
≥
3,
and
that
the
induction
hypothesis
is
in
force.
Let
(α
v
)
v∈Vert(G)
∈
Glu(Π
n
).
First,
let
us
observe
that
it
follows
from
the
in-
duction
hypothesis
that
there
exists
an
element
α
n−1
∈
Out
FC
(Π
n−1
)
brch
such
that
ρ
brch
n−1
(α
n−1
)
coincides
with
the
element
of
Glu(Π
n−1
)
deter-
n−1
be
an
mined
by
(α
v
)
v∈Vert(G)
∈
Glu(Π
n
)
[cf.
assertion
(i)].
Let
α
automorphism
of
Π
n−1
that
lifts
α
n−1
.
Write
α
n−1/n−2
for
the
outomor-
phism
of
Π
n−1/n−2
determined
by
α
n−1
and
α
n−2
for
the
automorphism
n−1
.
of
Π
n−2
determined
by
α
Next,
let
us
observe
that
one
verifies
easily
from
the
various
defi-
nitions
involved
that
Π
n/n−2
⊆
Π
n
may
be
regarded
as
the
“Π
2
”
as-
sociated
to
some
stable
log
curve
“X
log
”
over
(Spec
k)
log
.
Moreover,
this
stable
log
curve
may
be
taken
to
be
a
geometric
fiber
of
the
sort
discussed
in
Definition
3.1,
(iii),
in
the
case
of
the
projection
p
log
n−1/n−2
,
relative
to
a
point
“x
∈
X
n
(k)”
that
maps
to
the
interior
of
the
same
irreducible
component
of
X
log
,
relative
to
the
n
projections
to
X
log
.
In
particular,
by
fixing
such
a
stable
log
curve,
together
with
a
suitable
choice
of
lifting
α
n−1
[cf.
Theorem
4.7],
it
makes
sense
to
speak
of
Glu(Π
n/n−2
).
Moreover,
it
follows
immediately
from
our
choice
of
“x”
that
every
configuration
space
subgroup
that
appears
in
the
definition
[cf.
Definition
4.9,
(ii)]
of
Glu(Π
n/n−2
)
either
•
occurs
as
the
intersection
with
Π
n/n−2
of
some
configuration
space
subgroup
that
appears
in
the
definition
[cf.
Definition
4.9,
(iii)]
of
Glu(Π
n
)
or
•
projects
isomorphically,
via
the
projection
Π
n
→
Π
2
to
the
factors
labeled
n
and
n
−
1,
to
a
configuration
space
subgroup
of
Π
2
,
i.e.,
a
configuration
space
subgroup
that
appears
in
the
definition
[cf.
Definition
4.9,
(ii)]
of
Glu(Π
2
).
In
particular,
every
tripod
that
appears
in
the
definition
[cf.
Defini-
tion
4.9,
(ii)]
of
Glu(Π
n/n−2
)
occurs
as
a
tripod
of
a
configuration
space
subgroup
that
appears
either
in
the
definition
[cf.
Definition
4.9,
(iii)]
of
Glu(Π
n
)
or
in
the
definition
[cf.
Definition
4.9,
(ii)]
of
Glu(Π
2
).
Moreover,
it
follows
from
Theorem
4.7;
Lemma
3.2,
(iv);
Lemma
4.8,
(i),
that
the
various
α
v
’s
preserve
the
conjugacy
classes
of
these
config-
uration
space
subgroups
and
tripods
—
as
well
as
each
conjugacy
class
of
cuspidal
inertia
subgroups
of
each
of
these
tripods!
—
that
appear
in
the
definition
[cf.
Definition
4.9,
(ii)]
of
Glu(Π
n/n−2
).
Thus,
we
con-
clude
from
Theorem
3.18,
(ii),
together
with
Definition
4.9,
(iii),
in
the
case
of
Glu(Π
n
),
and
Definition
4.9,
(ii),
in
the
case
of
Glu(Π
2
),
that
136
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(α
v
)
v∈Vert(G)
determines
an
element
∈
Glu(Π
n/n−2
),
hence,
by
Claim
4.14.B,
an
element
α
n/n−2
∈
Out
FC
(Π
n/n−2
)
that
lifts
the
element
α
n−1/n−2
∈
Out(Π
n−1/n−2
).
Now
we
claim
that
the
following
assertion
holds:
Claim
4.14.C:
This
outomorphism
α
n/n−2
of
Π
n/n−2
is
compatible
with
the
automorphism
α
n−2
of
Π
n−2
rel-
ative
to
the
homomorphism
Π
n−2
→
Out(Π
n/n−2
)
in-
duced
by
the
natural
exact
sequence
of
profinite
groups
p
Π
n/n−2
1
−→
Π
n−2/n
−→
Π
n
−→
Π
n−2
−→
1
.
Indeed,
this
follows
immediately
from
the
corresponding
fact
for
α
n−1/n−2
[which
follows
from
the
existence
of
α
n−1
],
together
with
the
injectivity
FC
of
the
natural
homomorphism
Out
(Π
n/n−2
)
→
Out
FC
(Π
n−1/n−2
)
[cf.
[NodNon],
Theorem
B].
This
completes
the
proof
of
Claim
4.14.C.
∼
Thus,
by
applying
Claim
4.14.C
and
the
natural
isomorphism
Π
n
→
out
Π
n/n−2
Π
n−2
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0],
we
obtain
an
outomorphism
α
n
of
Π
n
that
lifts
the
outo-
morphism
α
n−1
of
Π
n−1
.
Thus,
it
follows
immediately
from
Lemma
4.10,
(i),
that
ρ
brch
n
(α
n
)
=
(α
v
)
v∈Vert(G)
.
This
completes
the
proof
of
the
sur-
jectivity
of
ρ
brch
n
,
hence
also
of
assertion
(iii).
Remark
4.14.1.
In
the
notation
of
Theorem
4.14,
observe
that
the
data
of
collections
of
smooth
log
curves
that
[by
gluing
at
prescribed
cusps]
give
rise
to
a
stable
log
curve
whose
associated
semi-graph
of
anabelioids
[of
pro-Σ
PSC-type]
is
isomorphic
to
G
form
a
smooth,
connected
moduli
stack.
In
particular,
by
considering
a
suitable
path
in
the
étale
fundamental
groupoid
of
this
moduli
stack,
one
verifies
immediately
that
one
may
reduce
the
verification
of
an
“isomorphism
version”
—
i.e.,
concerning
PFC-admissible
[cf.
[CbTpI],
Definition
1.4,
(iii)]
outer
isomorphisms
between
the
pro-Σ
fundamental
groups
of
the
configuration
spaces
associated
to
two
a
priori
distinct
stable
log
curves
“X
log
”
and
“Y
log
”
—
of
Theorem
4.14
to
the
“automor-
phism
version”
given
in
Theorem
4.14
[cf.
[CmbCsp],
Remark
4.1.4].
A
similar
statement
may
be
made
concerning
Theorem
4.7.
We
leave
the
routine
details
to
the
interested
reader.
In
the
present
monograph,
we
restricted
our
attention
to
the
“automorphism
versions”
of
these
results
in
order
to
simplify
the
[already
somewhat
complicated!]
nota-
tion.
Remark
4.14.2.
One
may
regard
[CmbCsp],
Corollary
3.3,
as
a
special
discussed
in
Theorem
4.14,
i.e.,
the
case
case
of
the
surjectivity
of
ρ
brch
n
COMBINATORIAL
ANABELIAN
TOPICS
II
137
in
which
X
log
is
obtained
by
gluing
a
tripod
to
a
smooth
log
curve
along
a
cusp
of
the
smooth
log
curve.
Corollary
4.15
(Surjectivity
result).
In
the
notation
of
Theorem
3.16,
suppose
that
n
≥
3.
If
r
=
0,
then
we
suppose
further
that
n
≥
4.
Then
the
tripod
homomorphism
T
Π
tpd
:
Out
F
(Π
n
)
−→
Out
C
(Π
tpd
)
Δ+
[cf.
Definition
3.19]
is
surjective.
Proof.
Let
α
∈
Out
C
(Π
tpd
)
Δ+
.
First,
let
us
observe
that
—
by
consid-
ering
a
suitable
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
and
ap-
plying
a
suitable
specialization
isomorphism
[cf.
Proposition
3.24,
(i);
the
discussion
preceding
[CmbCsp],
Definition
2.1,
as
well
as
[CbTpI],
Remark
5.6.1]
—
to
verify
Corollary
4.15,
we
may
assume
without
loss
of
generality
that
G
is
totally
degenerate
[cf.
[CbTpI],
Definition
2.3,
(iv)],
i.e.,
that
every
vertex
of
G
is
a
tripod
of
X
n
log
[cf.
Defini-
tion
3.1,
(v)].
Then
since
α
∈
Out
C
(Π
tpd
)
Δ+
,
it
follows
immediately
from
[CmbCsp],
Corollary
4.2,
(i),
(ii)
[cf.
also
[CmbCsp],
Definition
1.11,
(i)],
that
there
exists
an
element
α
n
∈
Out
FC
(Π
tpd
n
)
—
where
we
for
the
“Π
”
that
occurs
in
the
case
where
we
take
“X
log
”
write
Π
tpd
n
n
to
be
a
tripod
—
such
that
α
arises
as
the
image
of
α
n
via
the
nat-
FC
tpd
ural
injection
Out
FC
(Π
tpd
)
of
[NodNon],
Theorem
B.
n
)
→
Out
(Π
Thus,
it
follows
immediately
from
Theorem
4.14,
(iii),
that
there
ex-
—
the
ists
an
element
β
∈
Out
FC
(Π
n
)
brch
that
lifts
—
relative
to
ρ
brch
n
element
of
Glu(Π
n
)
[cf.
Theorems
3.16,
(v);
3.18,
(ii)]
determined
by
α
n
∈
Out
FC
(Π
tpd
n
).
[Here,
recall
that
we
have
assumed
that
G
is
totally
degenerate.]
Finally,
it
follows
from
Theorems
3.16,
(v);
3.18,
(ii),
that
T
Π
tpd
(β)
=
α,
i.e.,
that
α
is
contained
in
the
image
of
T
Π
tpd
.
This
completes
the
proof
of
Corollary
4.15.
Corollary
4.16
(Absolute
anabelian
cuspidalization
for
stable
log
curves
over
finite
fields).
Let
p,
l
X
,
l
Y
be
prime
numbers
such
that
p
∈
{l
X
,
l
Y
};
(g
X
,
r
X
),
(g
Y
,
r
Y
)
pairs
of
nonnegative
integers
such
that
2g
X
−2+r
X
,
2g
Y
−2+r
Y
>
0;
k
X
,
k
Y
finite
fields
of
characteristic
p;
k
X
,
k
Y
algebraic
closures
of
k
X
,
k
Y
;
(Spec
k
X
)
log
,
(Spec
k
Y
)
log
the
log
schemes
obtained
by
equipping
Spec
k
X
,
Spec
k
Y
with
the
log
structures
determined
by
the
fs
charts
N
→
k
X
,
N
→
k
Y
that
map
1
→
0;
X
log
,
Y
log
stable
log
curves
of
type
(g
X
,
r
X
),
(g
Y
,
r
Y
)
over
(Spec
k
X
)
log
,
(Spec
k
Y
)
log
;
def
def
def
def
log
G
log
k
X
=
π
1
((Spec
k
X
)
)
G
k
X
=
Gal(k
X
/k
X
)
,
log
G
log
k
Y
=
π
1
((Spec
k
Y
)
)
G
k
Y
=
Gal(k
Y
/k
Y
)
138
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
the
natural
surjections
[well-defined
up
to
composition
with
an
inner
log
automorphism];
s
X
:
G
k
X
→
G
log
k
X
,
s
Y
:
G
k
Y
→
G
k
Y
sections
of
the
log
above
natural
surjections
G
log
k
X
G
k
X
,
G
k
Y
G
k
Y
.
For
each
positive
integer
n,
write
X
n
log
,
Y
n
log
for
the
n-th
log
configuration
spaces
[cf.
the
discussion
entitled
“Curves”
in
“Notations
and
Conventions”]
of
X
log
,
Y
log
;
X
Π
n
,
Y
Π
n
for
the
maximal
pro-l
X
,
pro-l
Y
quotients
of
the
log
log
kernels
of
the
natural
surjections
π
1
(X
n
log
)
G
log
k
X
,
π
1
(Y
n
)
G
k
Y
.
Then
the
sections
s
X
,
s
Y
determine
outer
actions
of
G
k
X
,
G
k
Y
on
X
Π
n
,
Y
Π
n
.
Thus,
we
obtain
profinite
groups
out
out
Π
n
s
X
G
k
X
,
Y
Π
n
s
Y
G
k
Y
X
[cf.
[MzTa],
Proposition
2.2,
(ii);
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
Let
out
out
∼
α
1
:
X
Π
1
s
X
G
k
X
−→
Y
Π
1
s
Y
G
k
Y
be
an
isomorphism
of
profinite
groups.
Then
l
X
=
l
Y
;
there
exists
a
unique
collection
of
isomorphisms
of
profinite
groups
out
out
∼
α
n
:
X
Π
n
s
X
G
k
X
−→
Y
Π
n
s
Y
G
k
Y
n≥1
out
—
well-defined
up
to
composition
with
an
inner
automorphism
of
Y
Π
n
s
Y
G
k
Y
by
an
element
of
the
intersection
Y
Ξ
n
⊆
Y
Π
n
of
the
fiber
subgroups
of
Y
Π
n
of
co-length
1
[cf.
[CmbCsp],
Definition
1.1,
(iii)]
—
such
that
each
diagram
out
X
α
n+1
out
Π
n+1
s
X
G
k
X
−−−→
Y
Π
n+1
s
Y
G
k
Y
⏐
⏐
⏐
⏐
X
out
Π
n
s
X
G
k
X
α
n
−−−
→
Y
out
Π
n
s
Y
G
k
Y
—
where
the
vertical
arrows
are
the
surjections
induced
by
the
pro-
log
log
jections
X
n+1
→
X
n
log
,
Y
n+1
→
Y
n
log
obtained
by
forgetting
the
factors
labeled
j,
for
some
j
∈
{1,
·
·
·
,
n+1}
—
commutes,
up
to
composition
with
a
Y
Ξ
n
-inner
automorphism.
Proof.
First,
let
us
observe
that
it
follows
from
Corollary
4.18,
(ii),
def
below
that
l
X
=
l
Y
.
Write
l
=
l
X
=
l
Y
.
Moreover,
it
follows
from
Corollary
4.18,
(viii),
below
[i.e.,
in
the
case
where
condition
(viii-1)
out
is
satisfied]
that
α
1
maps
X
Π
1
⊆
X
Π
1
s
X
G
k
X
bijectively
onto
Y
Π
1
⊆
Y
out
Π
1
s
Y
G
k
Y
.
In
particular,
α
1
induces
isomorphisms
of
profinite
groups
∼
∼
α
1
Π
:
X
Π
1
−→
Y
Π
1
,
α
0
:
G
k
X
−→
G
k
Y
.
For
∈
{X,
Y
},
write
G
for
the
semi-graph
of
anabelioids
of
pro-l
PSC-type
determined
by
log
;
Π
G
for
the
[pro-l]
fundamental
group
COMBINATORIAL
ANABELIAN
TOPICS
II
139
(l)
of
G
;
G
k
⊆
G
k
for
the
maximal
pro-l
closed
subgroup
of
G
k
;
(
=l)
G
k
for
the
maximal
pro-prime-to-l
closed
subgroup
of
G
k
.
Thus,
we
have
a
natural
π
1
(
log
)-orbit,
i.e.,
relative
to
composition
with
au-
tomorphisms
induced
by
conjugation
by
elements
of
π
1
(
log
),
of
iso-
∼
∼
morphisms
Π
1
→
Π
G
;
fix
an
isomorphism
Π
1
→
Π
G
that
belongs
to
the
collection
of
isomorphisms
that
constitutes
this
π
1
(
log
)-orbit
of
isomorphisms.
Moreover,
since
G
k
is
isomorphic
to
Z
as
an
abstract
profinite
group,
we
have
a
natural
decomposition
(l)
(
=l)
∼
G
k
×
G
k
−→
G
k
.
Thus,
the
isomorphism
α
0
naturally
decomposes
into
a
pair
of
isomor-
phisms
(l)
(l)
∼
(l)
(
=l)
α
0
:
G
k
X
−→
G
k
Y
,
α
0
(
=l)
∼
(
=l)
:
G
k
X
−→
G
k
Y
.
Next,
let
us
observe
that
since
Π
1
is
topologically
finitely
generated
[cf.
[MzTa],
Proposition
2.2,
(ii)]
and
pro-l,
one
verifies
easily
that
[by
replacing
G
k
by
a
suitable
open
subgroup
and
applying
the
injectivity
portion
of
[NodNon],
Theorem
B,
together
with
[CmbGC],
Corollary
2.7,
(i)]
we
may
assume
without
loss
of
generality
that
the
outer
action
of
G
k
on
Π
1
—
hence
[cf.
the
injectivity
portion
of
[NodNon],
Theo-
rem
B]
also
on
Π
n
for
each
positive
integer
n
—
factors
through
the
∼
(l)
(
=l)
(l)
quotient
G
k
←
G
k
×
G
k
G
k
.
Next,
let
us
recall
the
following
well-known
Facts:
(1)
Some
positive
tensor
power
of
the
l-adic
cyclotomic
character
of
G
k
factors
through
the
outer
action
of
G
k
on
Π
1
[cf.
Corollary
4.18,
(vii),
below].
(l)
(2)
The
restriction
to
G
k
⊆
G
k
of
any
positive
tensor
power
of
the
l-adic
cyclotomic
character
of
G
k
is
injective.
Thus,
it
follows
from
Facts
(1),
(2),
that
(3)
the
resulting
outer
action
of
G
k
on
Π
1
—
hence
also
on
Π
n
for
each
positive
integer
n
—
is
injective.
(l)
In
particular,
it
follows
immediately
from
the
slimness
of
Π
n
[cf.
[MzTa],
Proposition
2.2,
(ii)]
that
the
composite
Z
out
out
Π
n
s
G
k
(
Π
n
)
→
Π
n
s
G
k
G
k
determines
an
isomorphism
Z
∼
out
Π
n
s
G
k
(
Π
n
)
−→
G
=l
k
.
140
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
Thus,
if
we
identify
Z
out
Π
n
s
G
k
(
Π
n
)
with
G
=l
k
by
means
of
this
iso-
morphism,
then
we
obtain
a
natural
isomorphism
out
out
(l)
(
=l)
∼
Π
n
s
G
k
×
G
k
−→
Π
n
s
G
k
.
Next,
let
us
observe
that
the
following
assertion
holds:
Claim
4.16.A:
There
exists
a
positive
power
q
of
p
such
that
log
p
(q)
is
divisible
by
log
p
(#k
X
),
log
p
(#k
Y
),
and,
moreover,
(l)
(l)
(l)
α
0
((Fr
q
)
k
X
)
=
(Fr
q
)
k
Y
—
where
we
write
(Fr
q
)
k
X
∈
G
k
X
,
(Fr
q
)
k
Y
∈
G
k
Y
for
(l)
the
q-power
Frobenius
elements
of
G
k
X
,
G
k
Y
;
(Fr
q
)
k
X
∈
(l)
(l)
(l)
G
k
X
,
(Fr
q
)
k
Y
∈
G
k
Y
for
the
respective
images
of
(Fr
q
)
k
X
∈
(l)
(l)
G
k
X
,
(Fr
q
)
k
Y
∈
G
k
Y
in
G
k
X
,
G
k
Y
.
Indeed,
this
follows
immediately
from
Corollary
4.18,
(vii),
below,
to-
gether
with
Fact
(2).
Write
H
k
X
⊆
G
k
X
,
H
k
Y
⊆
G
k
Y
for
the
open
subgroups
of
G
k
X
,
G
k
Y
topologically
generated
by
(Fr
q
)
k
X
∈
G
k
X
,
(Fr
q
)
k
Y
∈
G
k
Y
[cf.
Claim
4.16.A];
U
k
Y
⊆
G
k
Y
for
the
open
subgroup
of
G
k
Y
topologically
gener-
(l)
(l)
ated
by
α
0
((Fr
q
)
k
X
)
∈
G
k
Y
;
H
k
X
⊆
G
k
X
for
the
image
of
H
k
X
⊆
G
k
X
in
(l)
(l)
(l)
(l)
(l)
G
k
X
;
H
k
Y
,
U
k
Y
⊆
G
k
Y
for
the
images
of
H
k
Y
,
U
k
Y
⊆
G
k
Y
in
G
k
Y
.
Then
(l)
(l)
it
follows
from
Claim
4.16.A
that
we
have
an
equality
H
k
Y
=
U
k
Y
,
and,
∼
moreover,
that
the
isomorphism
H
k
X
→
U
k
Y
induced
by
α
0
induces
an
(l)
∼
(l)
(l)
isomorphism
H
k
X
→
U
k
Y
=
H
k
Y
.
In
particular,
one
verifies
easily
that
∼
there
exists
an
isomorphism
of
profinite
groups
α
0
H
:
H
k
X
→
H
k
Y
that
(a)
maps
(Fr
q
)
k
X
∈
G
k
X
to
(Fr
q
)
k
Y
∈
G
k
Y
,
which
thus
implies
that
∼
(b)
the
isomorphism
H
k
X
→
H
k
Y
induced
by
α
0
H
coincides
with
(l)
∼
(l)
(l)
the
above
isomorphism
H
k
X
→
U
k
Y
=
H
k
Y
induced
by
α
0
.
(l)
(l)
Moreover,
it
follows
immediately
from
(b),
together
with
the
existence
of
the
natural
isomorphisms
out
out
(l)
(
=l)
∼
X
Π
n
s
X
G
k
X
×
G
k
X
−→
X
Π
n
s
X
G
k
X
,
out
out
(l)
(
=l)
∼
Y
Π
n
s
Y
G
k
Y
×
G
k
Y
−→
Y
Π
n
s
Y
G
k
Y
[cf.
the
discussion
preceding
Claim
4.16.A],
that
there
exists
an
iso-
morphism
out
∼
out
α
1
H
:
X
Π
1
s
X
H
k
X
−→
Y
Π
1
s
Y
H
k
Y
such
that
COMBINATORIAL
ANABELIAN
TOPICS
II
141
(c)
the
isomorphism
“α
0
”
of
H
k
X
with
H
k
Y
that
occurs
in
the
case
where
we
take
the
“α
1
”
to
be
α
1
H
coincides
with
α
0
H
[i.e.,
roughly
speaking,
α
1
H
lies
over
α
0
H
],
and,
moreover,
(d)
the
isomorphism
“α
1
Π
”
of
X
Π
1
with
Y
Π
1
that
occurs
in
the
case
where
we
take
the
“α
1
”
to
be
α
1
H
coincides
with
[the
original]
α
1
Π
[i.e.,
roughly
speaking,
α
1
H
restricts
to
α
1
Π
on
X
Π
1
].
In
particular,
we
conclude,
again
by
the
existence
of
the
natural
iso-
morphisms
out
out
(l)
(
=l)
∼
X
Π
n
s
X
G
k
X
×
G
k
X
−→
X
Π
n
s
X
G
k
X
,
out
out
(l)
(
=l)
∼
Y
Π
n
s
Y
G
k
Y
×
G
k
Y
−→
Y
Π
n
s
Y
G
k
Y
,
together
with
the
injectivity
portion
of
[NodNon],
Theorem
B,
and
[CmbGC],
Corollary
2.7,
(i),
that,
to
verify
Corollary
4.16
—
by
re-
placing
G
k
X
,
G
k
Y
,
α
1
by
H
k
X
,
H
k
Y
,
α
1
H
—
we
may
assume
without
loss
of
generality
that
#k
X
=
#k
Y
,
and
that
α
0
maps
the
#k
X
-power
Frobenius
element
of
G
k
X
to
the
#k
Y
-power
Frobenius
element
of
G
k
Y
.
We
may
also
assume
without
loss
of
generality
—
by
replacing
G
k
,
where
∈
{X,
Y
},
by
a
suitable
open
subgroup
of
G
k
if
necessary
—
that
the
following
condition
holds:
(e)
for
∈
{X,
Y
},
G
k
acts
trivially
on
the
underlying
semi-
graph
of
G
.
Next,
let
us
observe
that
the
uniqueness
portion
of
Corollary
4.16
follows
immediately
from
the
injectivity
portion
of
[NodNon],
Theorem
B,
and
[CmbGC],
Corollary
2.7,
(i).
Thus,
it
remains
to
verify
the
existence
of
a
collection
of
α
n
’s
as
in
the
statement
of
Corollary
4.16.
To
this
end,
for
each
positive
integer
i,
∈
{X,
Y
},
and
v
∈
Vert(G
),
write
(
Π
v
)
i
⊆
Π
i
for
the
configuration
space
subgroup
of
Π
i
as-
sociated
to
v
∈
Vert(G
)
[well-defined
up
to
Π
i
-conjugation
—
cf.
Definition
4.3].
Next,
let
us
observe
that
∼
(f)
the
isomorphism
Π
G
X
→
Π
G
Y
determined
by
α
1
Π
and
the
fixed
∼
∼
isomorphisms
X
Π
1
→
Π
G
X
,
Y
Π
1
→
Π
G
Y
is
graphic
[cf.
Corol-
lary
4.18,
(iii),
(iv),
below].
∼
Write
α
Vert
:
Vert(G
X
)
→
Vert(G
Y
)
for
the
bijection
determined
by
∼
the
graphic
isomorphism
Π
G
X
→
Π
G
Y
of
(f).
Thus,
for
each
v
∈
∼
Vert(G
X
),
the
isomorphism
Π
G
X
→
Π
G
Y
of
(f)
determines
an
outer
iso-
∼
morphism
β
v
:
(
X
Π
v
)
1
→
(
Y
Π
α
Vert
(v)
)
1
[cf.
[CmbGC],
Proposition
1.2,
(ii);
[CbTpI],
Lemma
2.12,
(i),
(ii),
(iii)],
which
is
compatible
with
the
respective
natural
outer
actions
of
G
k
X
,
G
k
Y
[cf.
(e)].
In
par-
ticular,
by
applying
[Wkb],
Theorem
C,
to
this
outer
isomorphism
142
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
∼
β
v
:
(
X
Π
v
)
1
→
(
Y
Π
α
Vert
(v)
)
1
,
we
obtain
[cf.
[CmbGC],
Corollary
2.7,
(i)]
a
PFC-admissible
[cf.
[CbTpI],
Definition
1.4,
(iii)]
outomorphism
∼
β
v,n
:
(
X
Π
v
)
n
→
(
Y
Π
α
Vert
(v)
)
n
,
which
is
compatible
with
the
respective
natural
outer
actions
of
G
k
X
,
G
k
Y
[cf.
(e)].
Moreover,
since
the
β
v
’s
∼
arise
from
a
single
isomorphism
Π
G
X
→
Π
G
Y
,
one
verifies
immediately
from
[CbTpI],
Corollary
3.9,
(ii),
(v),
and
the
injectivity
discussed
in
[Hsh],
Remark
6,
(iv)
[i.e.,
applied
to
the
difference
between
the
vari-
ous
outer
isomorphisms,
determined
by
β
v,n
,
between
tripods
of
(
X
Π
v
)
n
and
tripods
of
(
Y
Π
α
Vert
(v)
)
n
],
that
the
collection
(β
v,n
)
v∈Vert(G
X
)
is
con-
tained
in
the
set
which
corresponds
—
in
the
“isomorphism
version”
of
Theorem
4.14
discussed
in
Remark
4.14.1
—
to
the
set
“Glu(Π
n
)”
in
the
statement
of
Theorem
4.14.
In
particular,
it
follows
from
the
“isomorphism
version”
of
Theorem
4.14,
(i),
(iii),
discussed
in
Re-
mark
4.14.1
that
the
outer
isomorphism
determined
by
the
isomor-
∼
phism
α
1
Π
:
X
Π
1
→
Y
Π
1
and
the
collection
(β
v,n
)
v∈Vert(G)
uniquely
deter-
∼
mine
a
PFC-admissible
outer
isomorphism
β
n
:
X
Π
n
→
Y
Π
n
which
—
by
the
injectivity
portion
of
[NodNon],
Theorem
B
—
is
compatible
with
the
respective
outer
actions
of
G
k
X
,
G
k
Y
.
Finally,
one
verifies
immedi-
ately
that
one
may
construct
a
collection
of
α
n
’s
as
in
the
statement
of
Corollary
4.16
from
the
collection
of
the
β
n
’s.
This
completes
the
proof
of
the
existence
of
α
n
’s,
hence
also
of
Corollary
4.16.
Remark
4.16.1.
Corollary
4.16
may
be
regarded
as
a
generalization
of
[AbsCsp],
Theorem
3.1;
[Hsh],
Theorem
0.1;
[Wkb],
Theorem
C.
Corollary
4.17
(Commensurator
of
the
image
of
the
absolute
Galois
group
of
a
finite
field
in
the
totally
degenerate
case).
Let
n
be
a
positive
integer;
p,
l
two
distinct
prime
numbers;
(g,
r)
a
pair
of
nonnegative
integers
=
(0,
3)
such
that
2g
−2+r
>
0;
k
a
finite
field
of
characteristic
p;
k
an
algebraic
closure
of
k;
(Spec
k)
log
the
log
scheme
obtained
by
equipping
Spec
k
with
the
log
structure
determined
by
the
fs
chart
N
→
k
that
maps
1
→
0;
X
log
a
stable
log
curve
of
type
(g,
r)
over
(Spec
k)
log
.
Write
G
for
the
semi-graph
of
anabelioids
of
pro-l
PSC-type
associated
to
the
stable
log
curve
X
log
;
G
for
the
underlying
semi-graph
of
G;
Π
G
for
the
[pro-l]
fundamental
group
of
G;
def
def
log
G
log
k
=
π
1
((Spec
k)
)
G
k
=
Gal(k/k)
for
the
natural
surjection
[well-defined
up
to
composition
with
an
inner
automorphism].
For
each
positive
integer
i,
write
X
i
log
for
the
i-th
log
configuration
space
[cf.
the
discussion
entitled
“Curves”
in
“No-
tations
and
Conventions”]
of
X
log
;
Π
i
for
the
maximal
pro-l
quotient
of
the
kernel
of
the
natural
surjection
π
1
(X
i
log
)
G
log
k
.
Thus,
we
COMBINATORIAL
ANABELIAN
TOPICS
II
143
have
a
natural
π
1
(X
log
)-orbit,
i.e.,
relative
to
composition
with
auto-
morphisms
induced
by
conjugation
by
elements
of
π
1
(X
log
),
of
isomor-
∼
phisms
Π
1
→
Π
G
and
a
natural
outer
action
FC
ρ
X
log
:
G
log
k
−→
Out
(Π
i
)
i
[cf.
the
notation
of
[CmbCsp],
Definition
1.1,
(ii)].
Fix
an
outer
∼
isomorphism
Π
1
→
Π
G
whose
constituent
isomorphisms
belong
to
the
above
π
1
(X
log
)-orbit
of
isomorphisms.
Let
H
⊆
G
log
k
be
a
closed
sub-
log
group
of
G
k
whose
image
in
G
k
is
open.
Write
I
H
⊆
H
for
the
kernel
of
the
composite
H
→
G
log
k
G
k
.
We
shall
say
that
H
is
of
l-Dehn
type
if
the
maximal
pro-l
quotient
of
I
H
is
nontrivial.
Suppose
that
the
stable
log
curve
X
log
is
totally
degenerate
[i.e.,
that
the
com-
plement
in
X
of
the
nodes
and
cusps
is
a
disjoint
union
of
tripods].
Then
the
following
hold:
(i)
The
image
ρ
X
log
(I
H
)
⊆
Out(Π
1
)
is
contained
in
Dehn(G)
⊆
1
∼
Out(Π
G
)
←
Out(Π
1
)
[cf.
the
notation
of
[CbTpI],
Definition
4.4].
Moreover,
the
image
ρ
X
log
(I
H
)
is
nontrivial
if
and
only
1
if
H
is
of
l-Dehn
type.
Write
C(ρ)
def
I
H
=
(ρ
X
log
(I
H
)
⊗
Z
l
Q
l
)
∩
Dehn(G)
⊆
Dehn(G)
1
[considered
in
Dehn(G)
⊗
Z
l
Q
l
—
cf.
[CbTpI],
Theorem
4.8,
(iv)].
(ii)
For
any
positive
integer
m
≤
n,
the
natural
injection
Out
FC
(Π
n
)
→
Out
FC
(Π
m
)
of
[NodNon],
Theorem
B,
induces
isomor-
phisms
∼
Z
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
−→
Z
Out
FC
(Π
m
)
(ρ
X
m
log
(H))
,
∼
loc
loc
Z
Out
FC
(Π
)
(ρ
X
log
(H))
−→
Z
Out
FC
(Π
)
(ρ
X
log
(H))
n
m
n
m
[cf.
the
discussion
entitled
“Topological
groups”
in
“Notations
and
Conventions”],
∼
N
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
−→
N
Out
FC
(Π
m
)
(ρ
X
m
log
(H))
,
∼
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
−→
C
Out
FC
(Π
m
)
(ρ
X
m
log
(H))
.
∼
(iii)
Relative
to
the
natural
inclusion
Aut(G)
(⊆
Out(Π
G
)
←
Out(Π
1
)),
the
following
equality
holds:
C
Out
FC
(Π
1
)
(ρ
X
log
(H))
=
C
Aut(G)
(ρ
X
log
(H))
.
1
1
In
particular,
we
have
natural
homomorphisms
of
profinite
groups
∼
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
→
C
Out
FC
(Π
1
)
(ρ
X
log
(H))
→
Aut(G)
,
1
χ
G
∼
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
→
C
Out
FC
(Π
1
)
(ρ
X
log
(H))
→
Z
∗
l
1
144
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
the
notation
of
[CbTpI],
Definition
3.8,
(ii)]
—
where
the
first
arrow
in
each
line
is
the
isomorphism
of
(ii).
By
abuse
of
notation
[i.e.,
since
ρ
X
n
log
(H)
is
not
necessarily
contained
in
Aut
|grph|
(G)
—
cf.
the
notation
of
[CbTpI],
Definition
2.6,
(i);
Remark
4.1.2
of
the
present
monograph],
write
Z
Aut
|grph|
(G)
(ρ
X
n
log
(H))
⊆
Z
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
,
loc
loc
Z
Aut
|grph|
(G)
(ρ
X
log
(H))
⊆
Z
Out
FC
(Π
)
(ρ
X
log
(H))
,
n
n
n
N
Aut
|grph|
(G)
(ρ
X
n
log
(H))
⊆
N
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
,
C
Aut
|grph|
(G)
(ρ
X
n
log
(H))
⊆
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
for
the
kernels
of
the
restrictions
of
the
composite
homomor-
phism
of
the
first
line
of
the
second
display
[of
the
present
(iii)]
to
loc
Z
Out
FC
(Π
n
)
(ρ
X
n
log
(H)),
Z
Out
FC
(Π
)
(ρ
X
log
(H))
,
n
n
N
Out
FC
(Π
n
)
(ρ
X
n
log
(H)),
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
,
respectively.
(iv)
Suppose
that
H
is
not
of
l-Dehn
type.
Then
we
have
equal-
ities
loc
Z
Aut
|grph|
(G)
(ρ
X
n
log
(H))
=
Z
Aut
|grph|
(G)
(ρ
X
log
(H))
n
=
N
Aut
|grph|
(G)
(ρ
X
n
log
(H))
=
C
Aut
|grph|
(G)
(ρ
X
n
log
(H))
[cf.
the
notation
of
(iii)].
Moreover,
each
of
the
four
groups
appearing
in
these
equalities
is,
in
fact,
independent
of
n
[cf.
(ii)].
(v)
Suppose
that
H
is
of
l-Dehn
type.
Then
the
composite
ho-
momorphism
of
the
first
line
of
the
second
display
of
(iii)
de-
termines
an
injection
of
profinite
groups
loc
Z
Out
FC
(Π
)
(ρ
X
log
(H))
→
Aut(G)
.
n
n
(vi)
Write
k
|grph|
(⊆
k)
for
the
[finite]
subfield
of
k
consisting
of
the
invariants
of
k
with
respect
to
[the
natural
action
on
k
of
]
the
kernel
of
the
natural
action
of
H
on
G.
Then
the
composite
homomorphism
of
the
second
line
of
the
second
display
of
(iii)
determines
natural
exact
sequences
of
profinite
groups
N
(ρ)
−→
N
Aut
|grph|
(G)
(ρ
X
n
log
(H))
−→
Z
∗
l
,
C(ρ)
−→
C
Aut
|grph|
(G)
(ρ
X
n
log
(H))
−→
Z
∗
l
1
−→
I
H
1
−→
I
H
[cf.
the
notation
of
(i),
(iii)]
—
where
ρ
X
n
log
(I
H
),
hence
also
N
(ρ)
def
(ρ
X
n
log
(I
H
)
⊆)
I
H
=
N
Aut
|grph|
(G)
(ρ
X
n
log
(H))
∩
Dehn(G)
COMBINATORIAL
ANABELIAN
TOPICS
II
145
C(ρ)
[cf.
(i),
(ii),
(iii)],
is
an
open
subgroup
of
I
H
;
the
image
of
the
third
arrow
in
each
line
contains
#k
|grph|
∈
Z
∗
l
and
does
not
depend
on
the
choice
of
n.
In
particular,
these
images
are
open;
if,
moreover,
#k
|grph|
∈
Z
∗
l
topologically
generates
Z
∗
l
,
then
the
third
arrows
in
each
line
are
surjective.
(vii)
The
closed
subgroup
ρ
X
n
log
(H)
⊆
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H)),
hence
also
N
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
(⊆
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H))),
is
open
in
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H)).
(viii)
Consider
the
following
conditions
[cf.
Remark
4.17.1
below]:
(1)
Write
Aut
(Spec
k)
log
(X
log
)
for
the
group
of
automorphisms
of
X
log
over
(Spec
k)
log
.
Then
the
natural
homomorphism
Aut
(Spec
k)
log
(X
log
)
−→
Aut(G)
is
surjective.
(2)
#k
|grph|
∈
Z
∗
l
topologically
generates
Z
∗
l
.
If
condition
(1)
is
satisfied,
and
H
is
of
l-Dehn
type,
then
we
have
an
equality
loc
Z
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
=
Z
Out
FC
(Π
)
(ρ
X
log
(H))
,
n
n
and,
moreover,
the
composite
homomorphism
of
the
first
line
of
the
second
display
of
(iii)
determines
an
isomorphism
∼
loc
Z
Out
FC
(Π
)
(ρ
X
log
(H))
−→
Aut(G)
.
n
n
If
conditions
(1)
and
(2)
are
satisfied,
then
the
composite
ho-
momorphisms
of
the
two
lines
of
the
second
display
of
(iii)
determine
natural
exact
sequences
of
profinite
groups
N
(ρ)
−→
N
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
−→
Aut(G)
×
Z
∗
l
−→
1
,
C(ρ)
−→
C
Out
FC
(Π
n
)
(ρ
X
n
log
(H))
−→
Aut(G)
×
Z
∗
l
−→
1
.
1
−→
I
H
1
−→
I
H
Proof.
Assertion
(i)
follows
immediately
from
the
various
definitions
involved,
together
with
[CbTpI],
Lemma
5.4,
(ii);
[CbTpI],
Proposi-
tion
5.6,
(ii).
Assertion
(ii)
follows
immediately
from
Corollary
4.16,
together
with
the
slimness
of
Π
i
for
each
positive
integer
i
[cf.
[MzTa],
Proposition
2.2,
(ii)]
and
the
openness
of
the
image
of
H
in
G
k
.
As-
sertion
(iii)
follows
immediately
from
[CmbGC],
Corollary
2.7,
(ii)
[cf.
also
the
proof
of
[CmbGC],
Proposition
2.4,
(v)],
together
with
the
openness
of
the
image
of
H
in
G
k
.
For
∈
{Z,
Z
loc
,
N,
C}
and
v
∈
Vert(G),
write
∼
def
=
Out
FC
(Π
1
)
(ρ
X
log
(H))
⊆
Out(Π
1
)
→
Out(Π
G
)
;
1
|grph|
=
∩
Aut
|grph|
(G)
⊆
Out(Π
G
)
def
146
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
[cf.
the
notation
of
[CbTpI],
Definition
2.6,
(i);
Remark
4.1.2
of
the
present
monograph];
pr
v
:
Aut
|grph|
(G)
−→
Aut
|grph|
(G|
v
)
for
the
homomorphism
determined
by
restriction
to
G|
v
[cf.
[CbTpI],
Definition
2.14,
(ii);
[CbTpI],
Remark
2.5.1,
(ii)];
v
⊆
Aut
|grph|
(G|
v
)
for
the
image
of
|grph|
⊆
Aut
|grph|
(G)
via
pr
v
.
Then
we
claim
that
the
following
assertion
holds:
Claim
4.17.A:
Let
v
∈
Vert(G).
Then
C
v
∩
Ker(χ
G|
v
)
=
{1}
[cf.
the
notation
of
[CbTpI],
Definition
3.8,
(ii)].
Indeed,
let
us
first
observe
that
since
Π
1
is
topologically
finitely
gen-
erated
[cf.
[MzTa],
Proposition
2.2,
(ii)]
and
pro-l,
one
verifies
easily
that
the
image
of
the
outer
action
ρ
X
log
admits
a
pro-l
open
subgroup.
1
Thus,
since
the
image
of
H
in
G
k
is
open,
it
follows
immediately
from
Corollary
4.18,
(vii),
below
that
C
v
⊆
Aut
|grph|
(G|
v
)
is
contained
in
the
local
centralizer
[cf.
the
discussion
entitled
“Topological
groups”
in
“No-
tations
and
Conventions”]
of
the
natural
image
of
G
k
in
Aut
|grph|
(G|
v
)
[cf.
the
fact
that
G|
v
is
of
type
(0,
3)].
Thus,
Claim
4.17.A
follows
im-
mediately
from
the
injectivity
discussed
in
[Hsh],
Remark
6,
(iv).
This
completes
the
proof
of
Claim
4.17.A.
Next,
we
claim
that
the
following
assertion
holds:
Claim
4.17.B:
Let
v
∈
Vert(G).
Then
C
|grph|
∩
Ker(pr
v
)
=
C
|grph|
∩
Ker(χ
G
)
=
C
|grph|
∩
Dehn(G)
;
loc
∩
Ker(pr
v
)
=
{1}
.
Z
|grph|
∩
Ker(pr
v
)
=
Z
|grph|
In
particular,
we
obtain
natural
isomorphisms
∼
∼
loc
Z
|grph|
−→
Z
v
,
Z
|grph|
−→
Z
v
loc
and
a
natural
exact
sequence
of
profinite
groups
χ
G
1
−→
C
|grph|
∩
Dehn(G)
−→
C
|grph|
−→
Z
∗
l
.
Indeed,
let
us
first
observe
that
the
equalities
of
the
first
line
of
the
first
display
of
Claim
4.17.B
follow
immediately
from
Claim
4.17.A,
together
with
[CbTpI],
Corollary
3.9,
(iv).
Moreover,
since
the
image
of
H
in
G
k
is
open,
the
equalities
of
the
second
line
of
the
first
display
of
Claim
4.17.B
follow
immediately
from
[CbTpI],
Theorem
4.8,
(iv),
(v),
together
with
the
equalities
of
the
first
line
of
the
first
display
of
Claim
4.17.B.
This
completes
the
proof
of
Claim
4.17.B.
Next,
we
verify
assertion
(iv).
Let
us
first
observe
that
it
follows
from
assertion
(ii)
that
it
suffices
to
verify
assertion
(iv)
in
the
case
where
n
=
1.
Next,
let
us
observe
that
it
follows
from
Lemma
3.9,
COMBINATORIAL
ANABELIAN
TOPICS
II
147
(ii),
that
C
|grph|
⊆
N
Out
FC
(Π
1
)
(Z
loc
),
which
thus
implies
that
we
have
a
loc
natural
action
[by
conjugation]
of
C
|grph|
on
Z
loc
,
hence
also
on
Z
|grph|
,
as
well
as
a
natural
[trivial!]
action
of
C
|grph|
on
Aut(G).
Moreover,
by
considering
the
inclusion
∼
loc
(C
|grph|
⊇)
Z
|grph|
→
Z
v
loc
→
Z
∗
l
induced
by
χ
G|
v
[cf.
Claims
4.17.A,
4.17.B],
we
conclude
that
the
ho-
momorphisms
of
the
two
lines
of
the
second
display
of
assertion
(iii)
determine
a
natural
[C
|grph|
-equivariant!]
injection
Z
loc
→
Aut(G)
×
Z
∗
l
.
Thus,
since
Z
∗
l
is
abelian,
it
follows
that
C
|grph|
acts
trivially
on
Z
loc
,
i.e.,
that
C
|grph|
⊆
Z
Out
FC
(Π
1
)
(Z
loc
).
On
the
other
hand,
since
H
is
not
of
l-Dehn
type,
one
verifies
easily
from
assertion
(i)
that
ρ
X
log
(H)
is
1
abelian,
hence
that
ρ
X
log
(H)
⊆
Z
⊆
Z
loc
.
Thus,
we
conclude
that
1
C
|grph|
⊆
Z
Out
FC
(Π
1
)
(Z
loc
)
∩
Aut
|grph|
(G)
⊆
Z
Out
FC
(Π
1
)
(ρ
X
log
(H))
∩
Aut
|grph|
(G)
1
=
Z
∩
Aut
|grph|
(G)
=
Z
|grph|
.
This
completes
the
proof
of
assertion
(iv).
Next,
we
verify
assertion
(v).
First,
let
us
observe
that
it
follows
from
assertion
(ii)
that,
to
verify
assertion
(v),
it
suffices
to
verify
that
loc
loc
Z
|grph|
=
{1},
hence,
by
Claim
4.17.B,
that
χ
G
(Z
|grph|
)
=
{1}.
On
the
other
hand,
since
H
is
of
l-Dehn
type,
by
considering
the
conjugation
loc
action
of
Z
|grph|
on
ρ
X
log
(I
H
)
[which
is
nontrivial
by
assertion
(i)],
we
1
loc
)
=
{1},
conclude
from
[CbTpI],
Theorem
4.8,
(iv),
(v),
that
χ
G
(Z
|grph|
as
desired.
This
completes
the
proof
of
assertion
(v).
Next,
we
verify
assertion
(vi).
First,
we
observe
that
it
follows
from
N
(ρ)
assertions
(ii),
(iii)
that
the
definition
of
I
H
is
indeed
independent
of
n
[as
the
notation
suggests!].
Next,
we
claim
that
the
following
assertion
holds:
Claim
4.17.C:
N
(ρ)
ρ
X
log
(I
H
)
⊆
N
|grph|
∩
Dehn(G)
=
I
H
1
C(ρ)
⊆
C
|grph|
∩
Dehn(G)
=
I
H
.
Indeed,
the
final
equality
follows
immediately
from
an
elementary
com-
putation
[in
which
we
apply
[CbTpI],
Theorem
4.8,
(iv),
(v)],
together
with
assertion
(i);
the
remainder
of
Claim
4.17.C
follows
immediately
from
the
various
definitions
involved,
together
with
assertion
(i).
This
completes
the
proof
of
Claim
4.17.C.
Now
it
follows
immediately
from
Claims
4.17.B,
4.17.C,
together
with
assertion
(ii),
that
the
composite
homomorphism
of
the
second
line
of
the
second
display
of
(iii)
deter-
mines
the
two
displayed
exact
sequences
of
assertion
(vi),
and
that
N
(ρ)
C(ρ)
ρ
X
log
(I
H
),
hence
also
I
H
,
is
an
open
subgroup
of
I
H
.
Moreover,
1
148
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
since
[it
is
immediate
that]
the
image,
via
ρ
X
n
log
,
of
the
kernel
of
the
natural
action
of
H
on
G
is
contained
in
N
|grph|
,
the
image
of
the
third
arrow
in
each
line
of
the
displayed
sequences
of
assertion
(vi)
contains
#k
|grph|
∈
Z
∗
l
.
Finally,
it
follows
from
assertion
(ii)
that
the
image
of
the
third
arrow
in
each
line
of
the
displayed
sequences
of
assertion
(vi)
does
not
depend
on
the
choice
of
n.
This
completes
the
proof
of
assertion
(vi).
Assertion
(vii)
follows
immediately
from
assertions
(iii)
and
(vi),
together
with
the
finiteness
of
Aut(G).
Assertion
(viii)
follows
im-
mediately
from
assertions
(v)
and
(vi).
This
completes
the
proof
of
Corollary
4.17.
Remark
4.17.1.
(i)
One
verifies
easily
that
condition
(1)
of
Corollary
4.17,
(viii),
holds
if,
for
instance,
k
=
k
|grph|
,
and,
moreover,
the
lengths
[cf.
[CbTpI],
Definition
5.3,
(ii)]
of
the
various
nodes
of
X
log
[whose
base-change
from
k
to
k
may
be
thought
of
as
the
special
fiber
stable
log
curve
of
[CbTpI],
Definition
5.3]
coincide.
(ii)
In
a
similar
vein,
one
verifies
easily
that
condition
(2)
of
Corol-
lary
4.17,
(viii),
holds
if,
for
instance,
k
|grph|
=
F
p
,
and,
more-
2
over,
p
remains
prime
in
the
cyclotomic
extension
Q(e
2πi/l
),
√
where
i
=
−1,
and
we
assume
that
l
is
odd.
Remark
4.17.2.
The
computation,
in
the
case
where
n
=
1,
of
the
cen-
tralizer
(respectively,
normalizer
and
commensurator)
in
Corollary
4.17,
(viii),
may
be
thought
of
as
a
sort
of
relative
geometrically
pro-l
(respectively,
[semi-]
absolute
geometrically
pro-l)
version
of
the
Grothendieck
Conjecture
for
totally
degenerate
stable
log
curves
over
finite
fields.
In
fact,
the
proofs
of
these
computations
of
Corol-
lary
4.17,
(viii),
in
the
case
where
n
=
1,
only
involve
the
theory
of
[CmbGC]
and
[CbTpI].
On
the
other
hand,
these
computations
of
Corollary
4.17,
(viii),
can
only
be
performed
under
certain
relatively
restrictive
conditions
[cf.
Remark
4.17.1].
It
is
precisely
for
this
reason
that
Corollary
4.17,
(ii),
which
may
be
thought
of
as
an
application
of
the
theory
of
the
present
monograph,
is
of
interest
in
the
context
of
these
computations
of
Corollary
4.17,
(viii).
Corollary
4.18
(Compatibility
with
geometric
subgroups).
Let
p,
l
X
,
l
Y
be
prime
numbers
such
that
p
∈
{l
X
,
l
Y
};
(g
X
,
r
X
),
(g
Y
,
r
Y
)
pairs
of
nonnegative
integers
such
that
2g
X
−
2
+
r
X
,
2g
Y
−
2
+
r
Y
>
0;
k
X
,
k
Y
finite
fields
of
characteristic
p;
k
X
,
k
Y
algebraic
closures
of
k
X
,
k
Y
;
(Spec
k
X
)
log
,
(Spec
k
Y
)
log
the
log
schemes
obtained
by
equipping
COMBINATORIAL
ANABELIAN
TOPICS
II
149
Spec
k
X
,
Spec
k
Y
with
the
log
structures
determined
by
the
fs
charts
N
→
k
X
,
N
→
k
Y
that
map
1
→
0;
X
log
,
Y
log
stable
log
curves
of
type
(g
X
,
r
X
),
(g
Y
,
r
Y
)
over
(Spec
k
X
)
log
,
(Spec
k
Y
)
log
;
def
def
def
def
log
G
log
k
X
=
π
1
((Spec
k
X
)
)
G
k
X
=
Gal(k
X
/k
X
)
,
log
G
log
k
Y
=
π
1
((Spec
k
Y
)
)
G
k
Y
=
Gal(k
Y
/k
Y
)
the
natural
surjections
[well-defined
up
to
composition
with
an
inner
log
log
log
Y
automorphism];
X
H
⊆
G
log
k
X
,
H
⊆
G
k
Y
closed
subgroups
of
G
k
X
,
G
k
Y
;
X
I
⊆
X
H,
Y
I
⊆
Y
H
the
kernels
of
the
composites
X
H
→
G
log
k
X
G
k
X
,
log
Y
X
Y
H
→
G
k
Y
G
k
Y
;
Π,
Π
the
maximal
pro-l
X
,
pro-l
Y
quotients
of
log
)
G
log
the
kernels
of
the
natural
surjections
π
1
(X
log
)
G
log
k
X
,
π
1
(Y
k
Y
;
G
X
,
G
Y
the
semi-graphs
of
anabelioids
of
pro-l
PSC-type
determined
by
X
log
,
Y
log
;
Π
G
X
,
Π
G
Y
the
[pro-l]
fundamental
groups
of
G
X
,
G
Y
[so
we
have
natural
π
1
(X
log
)-,
π
1
(Y
log
)-orbits
—
i.e.,
relative
to
composition
with
automorphisms
induced
by
conjugation
by
elements
of
π
1
(X
log
),
∼
∼
π
1
(Y
log
)
—
of
isomorphisms
X
Π
→
Π
G
X
,
Y
Π
→
Π
G
Y
].
Then
the
natural
log
X
Y
X
outer
actions
of
G
log
k
X
,
G
k
Y
on
Π,
Π
determine
outer
actions
of
I
⊆
X
H,
Y
I
⊆
Y
H
on
X
Π,
Y
Π.
Thus,
we
obtain
profinite
groups
X
out
out
Π
X
I
⊆
X
Π
X
H,
Y
out
out
Π
Y
I
⊆
Y
Π
Y
H
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
Sup-
pose
that,
for
each
∈
{X,
Y
},
one
of
the
following
two
conditions
is
satisfied:
(a)
The
equality
H
=
G
log
k
holds.
(b)
The
composite
H
→
G
log
k
G
k
is
an
isomorphism.
We
shall
refer
to
a
closed
subgroup
of
X
Π,
Y
Π
obtained
by
forming
the
image
—
by
the
inverse
of
an
element
of
the
π
1
(X
log
)-,
π
1
(Y
log
)-
∼
∼
orbits
of
isomorphisms
X
Π
→
Π
G
X
,
Y
Π
→
Π
G
Y
discussed
above
—
in
X
Π,
Y
Π
of
a
verticial
(respectively,
cuspidal;
nodal;
edge-like)
subgroup
of
Π
G
X
,
Π
G
Y
as
a
verticial
(respectively,
cuspidal;
nodal;
edge-
out
out
like)
subgroup
of
X
Π
X
H,
Y
Π
Y
H.
We
shall
refer
to
a
closed
out
out
subgroup
of
X
Π
X
I,
Y
Π
Y
I
obtained
by
forming
the
normalizer
out
out
out
out
in
X
Π
X
I,
Y
Π
Y
I
[i.e.,
as
opposed
to
X
Π
X
H,
Y
Π
Y
H]
of
out
a
verticial
(respectively,
cuspidal;
nodal;
edge-like)
subgroup
of
X
Π
X
out
H,
Y
Π
Y
H
as
a
verticial
(respectively,
cuspidal;
nodal;
edge-
out
out
like)
I-decomposition
subgroup
of
X
Π
X
H,
Y
Π
Y
H.
[In
particular,
for
each
∈
{X,
Y
},
it
follows
from
[CmbGC],
Proposition
1.2,
(ii),
that
if
H
satisfies
condition
(b)
—
which
thus
implies
that
out
Π
=
Π
I
—
then
it
holds
that
a
closed
subgroup
of
Π
=
150
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
out
Π
I
is
a
verticial
(respectively,
cuspidal;
nodal;
edge-like)
out
subgroup
of
Π
H
if
and
only
if
it
is
a
verticial
(respectively,
out
cuspidal;
nodal;
edge-like)
I-decomposition
subgroup
of
Π
H.]
Let
out
out
∼
α
:
X
Π
X
H
−→
Y
Π
Y
H
be
an
isomorphism
of
profinite
groups.
Then
the
following
hold:
(i)
It
holds
that
X
H
satisfies
condition
(a)
(respectively,
(b))
if
and
only
if
Y
H
satisfies
condition
(a)
(respectively,
(b)).
(ii)
The
equality
l
X
=
l
Y
holds.
(iii)
The
isomorphism
α
induces
a
bijection
between
the
set
of
out
verticial
I-decomposition
subgroups
of
X
Π
X
H
and
out
the
set
of
verticial
I-decomposition
subgroups
of
Y
Π
Y
H.
(iv)
The
isomorphism
α
induces
a
bijection
between
the
set
of
cuspidal
(respectively,
nodal;
edge-like)
I-decomposition
out
subgroups
of
X
Π
X
H
and
the
set
of
cuspidal
(respectively,
out
nodal;
edge-like)
I-decomposition
subgroups
of
Y
Π
Y
H.
(v)
The
isomorphism
α
restricts
to
an
isomorphism
out
(
X
Π
X
H
⊇)
X
out
∼
out
Π
X
I
−→
Y
Π
Y
I
out
(⊆
Y
Π
Y
H).
(vi)
There
exists
a
positive
integer
n
χ
such
that
the
diagram
X
out
Y
out
⊗nχ
X
Π
H
−−−→
⏐
⏐
α
X
χ
G
X
H
−−−→
Aut(G
X
)
−−−
→
Z
∗
l
X
Π
Y
H
−−−→
Y
H
−−−→
Aut(G
Y
)
−−
⊗n
−→
Z
∗
l
Y
χ
G
χ
Y
—
where
χ
G
X
,
χ
G
Y
are
as
in
[CbTpI],
Definition
3.8,
(ii),
and
the
right-hand
vertical
equality
is
the
equality
that
arises
from
the
equality
l
X
=
l
Y
of
(ii)
—
commutes.
(vii)
The
composite
of
the
upper
(respectively,
lower)
three
horizon-
tal
arrows
of
the
diagram
of
(vi)
coincides
with
the
composite
of
the
upper
(respectively,
lower)
three
horizontal
arrows
of
the
COMBINATORIAL
ANABELIAN
TOPICS
II
151
diagram
X
out
Y
out
⊗nχ
X
X
Π
H
−−−→
⏐
⏐
α
χ
k
X
H
−−−→
G
k
X
−−−
→
Z
∗
l
X
Π
Y
H
−−−→
Y
H
−−−→
G
k
Y
−−
⊗n
−→
Z
∗
l
Y
χ
k
χ
Y
—
where
the
integer
n
χ
is
the
positive
integer
of
(vi);
the
right-
hand
vertical
equality
is
the
equality
that
arises
from
the
equal-
ity
l
X
=
l
Y
of
(ii);
the
third
upper
(respectively,
lower)
horizon-
tal
arrow
is
the
n
χ
-th
power
of
the
l
X
-
(respectively,
l
Y
-)
adic
cyclotomic
character
χ
k
X
of
G
k
X
(respectively,
χ
k
Y
of
G
k
Y
).
In
particular,
the
diagram
of
the
preceding
display
commutes.
(viii)
Suppose
that
one
of
the
following
three
conditions
is
satisfied:
(viii-1)
Either
X
H
or
Y
H
satisfies
condition
(b).
(viii-2)
It
holds
that
0
∈
{r
X
,
r
Y
}.
(viii-3)
The
isomorphism
α
induces
a
bijection
between
the
set
out
of
cuspidal
subgroups
of
X
Π
X
H
and
the
set
of
cus-
out
pidal
subgroups
of
Y
Π
Y
H.
Then
the
isomorphism
α
restricts
to
an
isomorphism
out
(
X
Π
X
H
⊇)
X
∼
Π
−→
Y
Π
out
(⊆
Y
Π
Y
H).
Proof.
First,
we
verify
assertions
(i),
(ii).
Let
∈
{X,
Y
}.
Write
Z
(p
)
for
the
pro-prime-to-p
completion
of
the
ring
Z
of
rational
integers.
The
following
Facts
are
well-known:
(1)
The
profinite
group
G
k
is
isomorphic
to
Z
as
an
abstract
profinite
group.
(2)
The
kernel
of
the
natural
surjection
G
log
k
G
k
admits
a
nat-
(p
)
ural
structure
of
free
Z
-module
of
rank
1.
(3)
The
natural
action
by
conjugation
of
G
k
on
the
kernel
of
the
natural
surjection
G
log
k
G
k
is
given
by
the
cyclotomic
char-
acter
[cf.
(2)].
In
particular,
for
each
prime
number
q
=
p,
every
maximal
pro-q
subgroup
of
G
log
k
admits
a
natural
struc-
ture
of
extension
of
Z
q
by
Z
q
[cf.
(1),
(2)].
Moreover,
the
image
of
the
action
Z
q
→
Aut(Z
q
)
=
Z
∗
q
determined
by
such
an
extension
is
open.
Moreover,
let
us
recall
[cf.,
e.g.,
[AbsTpI],
Proposition
2.3,
(i)]
that
the
following
holds:
152
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(4)
The
pro-l
group
Π
is
nontrivial,
center-free,
and
elastic
[cf.
[AbsTpI],
Definition
1.1,
(ii)].
Thus,
we
conclude
that
H
satisfies
condition
(b)
if
and
only
if
the
set
of
prime
numbers
q
such
that
every
maximal
pro-q
subgroup
of
out
Π
H
is
nonabelian
is
of
cardinality
1.
Moreover,
the
prime
number
l
may
be
characterized
as
the
unique
prime
number
q
such
that
there
out
exists
a
maximal
pro-q
subgroup
of
Π
H
that
is
not
isomorphic
to
a
closed
subgroup
of
an
extension
of
Z
q
by
Z
q
.
This
completes
the
proofs
of
assertions
(i),
(ii).
In
the
remainder
of
the
proof
of
Corollary
4.18,
we
shall
write
def
l
=
l
X
=
l
Y
[cf.
assertion
(ii)].
out
Next,
we
verify
assertion
(iii).
For
∈
{X,
Y
}
and
J
⊆
Π
H
an
open
subgroup,
write
J
RTF
for
the
maximal
pro-RTF-quotient
of
the
profinite
group
J
[cf.
[AbsTpI],
def
Proposition
1.2,
(iv)];
Π
J
=
J
∩
Π
⊆
Π;
H
J
⊆
H
for
the
image
out
of
the
composite
J
→
Π
H
H
[so
we
have
a
commutative
diagram
of
profinite
groups
1
−−−→
Π
J
−−−→
⏐
⏐
J
⏐
⏐
out
1
−−−→
−−−→
H
J
−−−→
1
⏐
⏐
Π
−−−→
Π
H
−−−→
H
−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
the
natural
inclusions];
G
Jk
⊆
G
k
for
the
image
of
the
composite
out
J
→
Π
H
H
→
G
log
k
G
k
;
(
Π
J
)
comb
for
the
“combinatorial
quotient”
of
Π
J
,
i.e.,
the
quotient
of
Π
J
by
the
normal
closed
subgroup
normally
topologically
generated
by
the
closed
subgroups
of
Π
J
obtained
by
forming
the
intersections
of
Π
J
out
with
the
verticial
subgroups
of
Π
H.
Now
we
claim
that
the
following
assertion
holds:
out
Claim
4.18.A:
For
∈
{X,
Y
}
and
J
⊆
Π
H
an
open
subgroup,
the
quotient
of
J
RTF
by
the
image
of
the
normal
closed
subgroup
Π
J
⊆
J
in
J
RTF
is
G
Jk
.
Indeed,
this
assertion
follows
immediately
from
Facts
(1),
(2),
(3).
Next,
we
claim
that
the
following
assertion
holds:
COMBINATORIAL
ANABELIAN
TOPICS
II
153
out
Claim
4.18.B:
Let
∈
{X,
Y
},
J
⊆
Π
H
an
open
subgroup,
Q
a
torsion-free
abelian
profinite
group,
and
J
→
Q
a
homomorphism
of
profinite
groups.
Then
the
composite
Π
J
→
J
→
Q
factors
through
the
natural
surjection
Π
J
(
Π
J
)
comb
.
To
this
end,
let
us
first
observe
that
since
[it
is
well-known
that]
the
image,
in
Z
∗
l
,
of
the
l-adic
cyclotomic
character
of
G
k
is
open,
one
verifies
immediately
that
the
image
by
the
composite
Π
J
→
J
→
out
Q
of
any
edge-like
subgroup
of
Π
H
[i.e.,
any
intersection
of
out
Π
J
with
any
edge-like
subgroup
of
Π
H]
is
trivial
[cf.,
e.g.,
[CmbGC],
Remark
1.1.3].
In
a
similar
vein,
it
follows
immediately
from
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”
[cf.,
e.g.,
[Mumf],
pp.
190-191]
that
the
image
by
the
composite
Π
J
→
out
J
Q
of
any
verticial
subgroup
of
Π
H
[i.e.,
any
intersection
of
out
Π
J
with
any
verticial
subgroup
of
Π
H]
is
trivial.
This
completes
the
proof
of
Claim
4.18.B.
Next,
we
claim
that
the
following
assertion
holds:
out
Claim
4.18.C:
For
∈
{X,
Y
}
and
J
⊆
Π
H
an
open
subgroup,
the
natural
exact
sequence
1
→
J
Π
→
J
→
H
J
→
1
fits
into
a
commutative
diagram
of
profinite
groups
1
−−−→
Π
J
⏐
⏐
−−−→
J
⏐
⏐
−−−→
H
J
−−−→
1
⏐
⏐
(
Π
J
)
comb
−−−→
J
RTF
−−−→
G
Jk
−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
the
natural
surjections.
Indeed,
Claim
4.18.C
follows
immediately,
in
light
of
Claim
4.18.A,
by
applying
Claim
4.18.B
to
the
various
RTF-subgroups
of
J
[cf.
[AbsTpI],
Definition
1.1,
(i)].
Next,
we
claim
that
the
following
assertion
holds:
Claim
4.18.D:
For
∈
{X,
Y
},
there
exists
an
open
out
subgroup
J
0
⊆
Π
H
that
satisfies
the
following
condition:
For
J
⊆
J
0
an
arbitrary
open
subgroup,
there
exists
an
open
subgroup
J
H
†
⊆
H
J
such
that
if
def
we
write
J
†
=
J
×
H
J
J
H
†
,
then
the
corresponding
left-
†
hand
lower
horizontal
arrow
(
Π
J
)
comb
→
(J
†
)
RTF
of
the
diagram
of
Claim
4.18.C
is
injective.
154
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
out
To
this
end,
let
J
0
⊆
Π
H
be
an
open
subgroup
such
that,
for
every
open
subgroup
J
⊆
J
0
,
the
quotient
(
Π
J
)
comb
is
a
center-free
free
pro-l
group
[where
we
note
that
one
verifies
easily
[cf.
[CmbGC],
Remark
1.1.3]
that
such
a
J
0
always
exists].
Next,
let
us
observe
that,
to
verify
Claim
4.18.D,
we
may
assume
without
loss
of
gener-
ality,
by
replacing
H
J
by
a
suitable
open
subgroup
of
H
J
,
that
the
outer
action
of
J
on
(
Π
J
)
comb
by
conjugation
is
trivial
[where
we
note
that
one
verifies
easily
that
such
an
open
subgroup
of
H
J
always
ex-
ists].
Since,
as
discussed
above,
(
Π
J
)
comb
is
center-free,
if
one
writes
def
J
comb
=
J/Ker(
Π
J
(
Π
J
)
comb
),
then
this
triviality
implies
that
the
inclusions
(
Π
J
)
comb
→
J
comb
←
Z
J
comb
((
Π
J
)
comb
)
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
deter-
mine
an
isomorphism
∼
(
Π
J
)
comb
×
Z
J
comb
((
Π
J
)
comb
)
−→
J
comb
.
On
the
other
hand,
since
(
Π
J
)
comb
is
a
free
pro-l
group,
the
nat-
ural
surjection
(
Π
J
)
comb
((
Π
J
)
comb
)
RTF
is
an
isomorphism.
In
∼
particular,
the
composite
of
natural
homomorphisms
(
Π
J
)
comb
→
((
Π
J
)
comb
)
RTF
→
(J
comb
)
RTF
is
injective.
Thus,
since
the
natural
surjection
J
(J
comb
)
RTF
factors
through
J
RTF
,
Claim
4.18.D
follows
immediately.
This
completes
the
proof
of
Claim
4.18.D.
Next,
we
claim
that
the
following
assertion
holds:
out
Claim
4.18.E:
Let
∈
{X,
Y
}
and
A
⊆
Π
H
a
closed
subgroup.
Then
the
following
two
conditions
are
equivalent:
out
(E-1)
The
closed
subgroup
A
⊆
Π
H
is
contained
out
in
a
verticial
I-decomposition
subgroup
of
Π
H.
out
(E-2)
For
J
⊆
Π
H
an
arbitrary
open
subgroup,
the
composite
A
∩
J
→
J
J
RTF
is
trivial.
To
this
end,
let
us
first
observe
that
the
implication
(E-1)
⇒
(E-2)
follows
immediately
from
Claim
4.18.C,
together
with
Facts
(1),
(2),
(3).
On
the
other
hand,
by
applying
Claims
4.18.C,
4.18.D
to
the
out
various
open
subgroups
of
Π
H
for
each
∈
{X,
Y
},
one
verifies
immediately
from
Proposition
1.5
that
the
implication
(E-2)
⇒
(E-
1)
holds.
This
completes
the
proof
of
Claim
4.18.E.
On
the
other
hand,
since
any
inclusion
of
verticial
I-decomposition
subgroups
is
an
equality
[cf.
[CmbGC],
Proposition
1.2,
(i),
(ii)],
assertion
(iii)
follows
immediately
from
Claim
4.18.E.
This
completes
the
proof
of
assertion
(iii).
COMBINATORIAL
ANABELIAN
TOPICS
II
155
Next,
we
verify
assertion
(iv).
We
begin
the
proof
of
assertion
(iv)
with
the
following
claim:
Claim
4.18.F:
Let
∈
{X,
Y
}.
Suppose
that
log
is
a
smooth
log
curve
over
(Spec
k
)
log
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0].
Then
the
inclusions
out
out
Π
→
Π
I
←
Z(
Π
I)
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0]
determine
an
isomorphism
out
∼
out
Π
×
Z(
Π
I)
−→
Π
I.
out
out
Moreover,
the
composite
Z(
Π
I)
→
Π
I
I
is
an
isomorphism.
In
particular,
if
H
satisfies
out
condition
(a)
(respectively,
(b)),
then
Z(
Π
I)
admits
a
structure
of
free
Z
(p
)
-module
of
rank
1
(re-
spectively,
is
trivial)
[cf.
Fact
(2)].
Indeed,
since
[we
have
assumed
that]
log
is
a
smooth
log
curve
over
(Spec
k
)
log
,
this
assertion
follows
immediately
from
the
slimness
of
Π
[cf.
[CmbGC],
Remark
1.1.3],
together
with
the
various
definitions
involved.
Next,
let
us
observe
that
it
follows
from
[CmbGC],
Proposition
1.2,
(ii),
that,
(5)
for
each
∈
{X,
Y
},
if
A
is
a
VCN-subgroup
of
Π,
then
out
the
intersection
of
Π
with
the
normalizer,
in
Π
I,
of
A
coincides
with
A.
Moreover,
let
us
also
observe
that
it
follows
from
[NodNon],
Remark
2.4.2;
[NodNon],
Remark
2.7.1,
that,
(6)
for
each
∈
{X,
Y
},
any
inclusion
of
VCN-subgroups
of
Π
out
gives
rise
to
an
inclusion
of
the
normalizers,
in
Π
I,
of
the
respective
VCN-subgroups.
Next,
we
claim
that
the
following
assertion
holds:
Claim
4.18.G:
The
isomorphism
α
induces
a
bijection
between
the
set
of
edge-like
I-decomposition
subgroups
out
of
X
Π
X
H
and
the
set
of
edge-like
I-decomposition
out
subgroups
of
Y
Π
Y
H.
To
this
end,
let
us
first
observe
that
it
follows
immediately
—
in
light
of
Facts
(5),
(6)
—
from
assertion
(iii)
that,
to
verify
Claim
4.18.G,
we
out
may
assume
without
loss
of
generality
—
by
replacing
Π
H
by
out
the
normalizer,
in
Π
H,
of
a
verticial
I-decomposition
subgroup
156
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
out
of
Π
H
for
each
∈
{X,
Y
}
—
that
X
log
,
Y
log
are
smooth
log
curves
over
(Spec
k
X
)
log
,
(Spec
k
Y
)
log
,
and
that
the
isomorphism
α
out
out
out
restricts
to
an
isomorphism
of
X
Π
X
I
(⊆
X
Π
X
H)
with
Y
Π
Y
I
out
(⊆
Y
Π
Y
H).
Next,
let
us
observe
that
if
X
H,
hence
also
Y
H
[cf.
assertion
(i)],
satisfies
condition
(b),
then
since
[it
is
well-known
that]
the
image,
in
Z
∗
l
,
of
the
l-adic
cyclotomic
character
of
G
k
is
open
for
each
∈
{X,
Y
},
Claim
4.18.G
follows
immediately
from
[CmbGC],
Corollary
2.7,
(i).
Thus,
in
the
remainder
of
the
proof
of
Claim
4.18.G,
we
may
assume
without
loss
of
generality
that
X
H,
hence
also
Y
H
[cf.
assertion
(i)],
satisfies
condition
(a).
Then,
by
applying
a
similar
argument
to
the
argument
in
the
proof
of
Claim
4.18.G
in
the
case
where
X
H
satisfies
condition
(b)
to
the
isomorphism
out
out
∼
out
out
(
X
Π
X
H)/Z(
X
Π
X
I)
−→
(
Y
Π
Y
H)/Z(
Y
Π
Y
I)
induced
by
α
[cf.
Claim
4.18.F],
we
conclude
that
this
induced
iso-
morphism
determines
a
bijection
between
the
set
of
images
of
edge-like
out
out
out
subgroups
of
X
Π
X
H
in
the
quotient
(
X
Π
X
H)/Z(
X
Π
X
I)
and
out
the
set
of
images
of
edge-like
subgroups
of
Y
Π
Y
H
in
the
quotient
out
out
(
Y
Π
Y
H)/Z(
Y
Π
Y
I).
Now
let
us
observe
that
it
follows
imme-
diately
from
Claim
4.18.F
and
[CmbGC],
Proposition
1.2,
(ii),
that,
out
for
each
∈
{X,
Y
}
and
each
edge-like
subgroup
A
⊆
Π
H,
the
out
edge-like
I-decomposition
subgroup
of
Π
H
obtained
by
forming
out
the
normalizer
of
A
in
Π
I
coincides
with
the
inverse
image
by
out
out
out
the
natural
surjection
Π
H
(
Π
H)/Z(
Π
I)
of
the
out
out
image
of
A
in
(
Π
H)/Z(
Π
I).
Thus,
Claim
4.18.G
holds.
This
completes
the
proof
of
Claim
4.18.G.
On
the
other
hand,
assertion
(iv)
follows
—
in
light
of
Facts
(5),
(6)
—
from
assertion
(iii),
Claim
4.18.G,
and
[CmbGC],
Proposition
1.5,
(i).
This
completes
the
proof
of
assertion
(iv).
Next,
we
verify
assertions
(v),
(vi),
(vii).
First,
we
observe
that
assertion
(vii)
is
a
formal
consequence
of
assertion
(vi),
together
with
[AbsCsp],
Proposition
1.2,
(ii);
[CbTpI],
Corollary
3.9,
(ii),
(iii).
Now
out
suppose
that
there
is
no
nodal
subgroup
of
X
Π
X
H,
hence
also
[cf.
out
assertion
(iv)]
of
Y
Π
Y
H.
Then
assertion
(v)
follows
from
assertion
(iii).
Moreover,
by
considering,
for
each
∈
{X,
Y
},
the
cyclotome
obtained
by
applying
the
construction
of
“Λ
v
”
of
[CbTpI],
Definition
3.8,
(i),
to
the
collection
of
data
consisting
of
COMBINATORIAL
ANABELIAN
TOPICS
II
out
157
out
•
the
profinite
group
(
Π
I)/Z(
Π
I)
and
out
out
•
the
various
images
in
(
Π
I)/Z(
Π
I)
of
the
edge-like
out
I-decomposition
subgroups
of
Π
H,
one
verifies
immediately
from
assertions
(iv),
(v),
together
with
Claim
4.18.F,
that
assertion
(vi)
[i.e.,
in
the
case
where
one
takes
“n
χ
”
in
the
statement
of
assertion
(vi)
to
be
1],
hence
also
assertion
(vii),
holds.
Thus,
in
the
remainder
of
the
proofs
of
assertions
(v),
(vi),
(vii),
out
we
may
assume
without
loss
of
generality
that
both
X
Π
X
H
and
Y
out
Π
Y
H
have
a
nodal
subgroup.
Then
one
verifies
immediately
from
assertions
(iii),
(iv)
[cf.
also
Facts
(5),
(6)],
together
with
Lemma
4.19
below
[cf.
[NodNon],
Definition
2.4,
(i);
[NodNon],
Remark
2.4.2],
that
assertion
(vi),
hence
also
assertion
(vii),
holds.
On
the
other
hand,
for
each
∈
{X,
Y
},
we
conclude
from
Fact
(2)
in
the
proof
of
Corollary
4.16
that
(7)
if
we
write
out
out
(
Π
I)
(l)
⊆
Π
I
for
the
[unique
—
cf.
Fact
(2)]
maximal
pro-l
subgroup
of
out
out
out
Π
I,
then
the
closed
subgroup
(
Π
I)
(l)
⊆
(
Π
out
out
I
⊆)
Π
H
coincides
with
the
closed
subgroup
of
Π
H
obtained
by
forming
the
unique
maximal
pro-l
subgroup
of
the
kernel
of
the
composite
of
the
relevant
[i.e.,
upper
if
=
X;
lower
if
=
Y
]
three
horizontal
arrows
of
the
diagram
of
assertion
(vii).
Moreover,
we
also
conclude
immediately
from
Facts
(1),
(2),
(3)
that,
for
each
∈
{X,
Y
},
(8)
the
kernel
of
the
composite
out
out
out
out
out
Π
H
Π
H/(
Π
I)
(l)
(
Π
H/(
Π
I)
(l)
)
RTF
out
coincides
with
the
closed
subgroup
Π
I.
In
particular,
it
follows
from
assertion
(vii)
and
Facts
(7),
(8)
that
the
out
out
isomorphism
α
restricts
to
an
isomorphism
of
X
Π
X
I
(⊆
X
Π
X
H)
out
out
with
Y
Π
Y
I
(⊆
Y
Π
Y
H),
as
desired.
This
completes
the
proof
of
assertion
(v).
Finally,
we
verify
assertion
(viii).
If
condition
(viii-1)
is
satisfied,
out
then
since
[it
follows
from
assertion
(i)
that]
Π
=
Π
I
for
each
∈
{X,
Y
},
assertion
(viii)
follows
from
assertion
(v).
Thus,
in
the
158
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
remainder
of
the
proof
of
assertion
(viii),
we
suppose
that
both
X
H
and
Y
H
satisfy
condition
(a).
Next,
suppose
that
condition
(viii-2)
is
satisfied.
Then
it
follows
from
assertion
(iv)
that
(r
X
,
r
Y
)
=
(0,
0).
Write
(l)
I
⊆
I
for
the
[unique
—
cf.
Fact
(2)]
maximal
pro-l
subgroup
of
I.
Then
one
verifies
easily
that
one
may
naturally
regard
I
(l)
as
a
quotient
of
out
(
Π
I)
(l)
[cf.
Fact
(7)],
and,
moreover,
that
the
closed
subgroup
out
Π
of
Π
H
coincides
with
the
kernel
of
the
natural
surjection
out
(
Π
I)
(l)
I
(l)
.
In
particular,
it
follows
from
assertions
(v),
(vii)
that,
to
verify
assertion
(viii)
in
the
case
where
condition
(viii-2)
is
satisfied,
it
suffices
to
verify
the
following
assertion:
out
Claim
4.18.H:
Let
∈
{X,
Y
}
and
(
Π
I)
(l)
A
out
a
quotient
of
(
Π
I)
(l)
.
Then
it
holds
that
this
out
quotient
(
Π
I)
(l)
A
coincides
with
the
quo-
out
tient
(
Π
I)
(l)
I
(l)
if
and
only
if
the
following
three
conditions
are
satisfied:
(H-1)
The
profinite
group
A
is
isomorphic
to
Z
l
as
an
abstract
profinite
group.
out
(H-2)
The
kernel
of
the
surjection
(
Π
I)
(l)
A
out
is
normal
in
Π
H.
Thus,
the
outer
action
out
out
of
Π
H
on
(
Π
I)
(l)
by
conjugation
out
induces
an
action
[cf.
(H-1)]
of
Π
H
on
the
quotient
A.
Moreover,
the
resulting
character
out
ρ
A
:
Π
H
→
Aut(A)
=
Z
∗
l
[cf.
(H-1)]
is
out
out
trivial
on
Π
I
⊆
Π
H.
(H-3)
The
n
χ
-th
power
of
the
character
ρ
A
of
(H-2)
co-
incides
with
the
composite
of
the
relevant
[i.e.,
upper
if
=
X;
lower
if
=
Y
]
three
horizontal
arrows
of
the
diagram
of
assertion
(vii).
First,
let
us
observe
that
it
follows
from
Facts
(2),
(3)
that
the
quo-
out
tient
(
Π
I)
(l)
I
(l)
satisfies
the
three
conditions
in
the
state-
ment
of
Claim
4.18.H.
Next,
let
us
observe
that
it
follows
immedi-
ately
from
[CmbGC],
Propositions
1.3,
2.6,
that
if
a
given
quotient
out
(
Π
I)
(l)
A
satisfies
conditions
(H-1),
(H-2),
then
the
image
in
A
of
an
arbitrary
nodal
[or,
equivalently,
edge-like
—
cf.
the
equality
COMBINATORIAL
ANABELIAN
TOPICS
II
159
out
(r
X
,
r
Y
)
=
(0,
0)
discussed
above]
subgroup
of
Π
H
is
trivial.
Thus,
it
follows
immediately
from
the
“Riemann
hypothesis
for
abelian
varieties
over
finite
fields”
[cf.,
e.g.,
[Mumf],
pp.
190-191],
together
with
Fact
(3)
and
condition
(H-3),
that
Claim
4.18.H
holds.
This
com-
pletes
the
proof
of
Claim
4.18.H,
hence
also
of
assertion
(viii)
in
the
case
where
condition
(viii-2)
is
satisfied.
Finally,
one
may
verify
assertion
(viii)
in
the
case
where
condition
(viii-3)
is
satisfied
by
applying
assertion
(viii)
in
the
case
where
condi-
tion
(viii-2)
is
satisfied.
Indeed,
let
us
first
observe
that
it
follows
im-
mediately
from
Fact
(1)
[which
implies
that
G
k
X
,
G
k
Y
are
torsion-free]
out
that
we
may
assume
without
loss
of
generality,
by
replacing
Π
H
out
by
a
suitable
open
subgroup
of
Π
H
for
each
∈
{X,
Y
},
that
g
X
,
g
Y
≥
2.
Then
we
may
assume
without
loss
of
generality,
by
re-
out
out
placing
Π
H
by
the
quotient
of
Π
H
by
the
normal
closed
subgroup
normally
topologically
generated
by
the
cuspidal
subgroups
for
each
∈
{X,
Y
},
that
(r
X
,
r
Y
)
=
(0,
0)
[cf.
(viii-3)].
Thus,
it
fol-
lows
from
assertion
(viii)
in
the
case
where
condition
(viii-2)
is
satisfied
that
assertion
(viii)
holds.
This
completes
the
proof
of
assertion
(viii),
hence
also
of
Corollary
4.18.
Remark
4.18.1.
In
the
situation
of
Corollary
4.18,
(viii),
if
one
omits
the
assumption
that
one
of
the
conditions
(viii-1),
(viii-2),
and
(viii-3)
holds,
then
the
conclusion
of
Corollary
4.18,
(viii),
no
longer
holds
in
general.
Indeed:
(i)
First,
we
consider
the
case
of
a
smooth
log
curve
[cf.
the
dis-
cussion
entitled
“Curves”
in
[CbTpI],
§0].
In
the
situation
def
of
Corollary
4.18,
write
l
=
l
X
.
Let
T
log
be
a
tripod
over
(Spec
k
X
)
log
[cf.
the
discussion
entitled
“Curves”
in
[CbTpI],
§0]
such
that
the
natural
action
of
G
log
k
X
on
the
set
of
cusps
of
T
log
is
trivial.
Then,
by
taking
“
T
H”
to
be
G
log
k
X
,
we
obtain
a
out
profinite
group
T
Π
T
H.
In
the
remainder
of
the
discussion
of
the
present
(i),
out
we
construct
an
automorphism
of
T
Π
T
H
that
out
does
not
preserve
the
closed
subgroup
T
Π
⊆
T
Π
T
H.
out
out
Let
C
⊆
T
Π
T
H
be
a
cuspidal
subgroup
of
T
Π
T
H.
def
out
out
Write
Z
=
Z(
T
Π
T
I)
for
the
center
of
T
Π
T
I
and
out
I
C
⊆
T
Π
T
H
for
the
cuspidal
I-decomposition
subgroup
of
160
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
T
out
out
Π
T
H
obtained
by
forming
the
normalizer
in
T
Π
T
I
of
C.
Then
out
out
(i-a)
the
natural
inclusions
T
Π
→
T
Π
T
I
and
Z
→
T
Π
T
I
out
∼
determine
an
isomorphism
T
Π×
Z
→
T
Π
T
I
[cf.
Claim
4.18.F].
Moreover,
the
natural
inclusions
C
→
I
C
and
∼
Z
→
I
C
determine
an
isomorphism
C
×Z
→
I
C
[cf.
Claim
4.18.F;
[CmbGC],
Proposition
1.2,
(ii)].
Moreover,
it
is
well-known
that
the
following
assertions
hold:
(i-b)
The
unique
maximal
pro-l
subgroup
of
Z
admits
a
struc-
ture
of
free
Z
l
-module
of
rank
1
[cf.
Claim
4.18.F].
More-
over,
the
natural
action
of
G
k
X
on
this
unique
maximal
out
∼
pro-l
subgroup
of
Z
(=
{0}
×
Z
⊆
T
Π
ab
×
Z
→
(
Π
ab
I)
)
[cf.
(i-a)]
induced
by
the
natural
outer
action
of
out
G
k
X
on
T
Π
T
I
is
given
by
the
l-adic
cyclotomic
char-
acter
[cf.
Fact
(3)
in
the
proof
of
Corollary
4.18].
(i-c)
The
pro-l
group
T
Π
ab
admits
a
structure
of
free
Z
l
-module
of
rank
2.
Moreover,
the
natural
action
of
G
k
X
on
T
Π
ab
out
∼
(=
T
Π
ab
×
{0}
⊆
T
Π
ab
×
Z
→
(
Π
I)
ab
)
[cf.
(i-a)]
out
induced
by
the
natural
outer
action
of
G
k
X
on
T
Π
T
I
is
given
by
the
l-adic
cyclotomic
character.
Thus,
since
C
admits
a
structure
of
free
Z
l
-module
of
rank
1
[cf.
[CmbGC],
Remark
1.1.3],
there
exists
a
nontrivial
homo-
morphism
φ
:
T
Π
(
T
Π
ab
)
→
Z
whose
kernel
is
topologically
normally
generated
by
C.
Now
write
α
I
for
the
automorphism
out
∼
of
the
profinite
group
T
Π
×
Z
(
→
T
Π
T
I)
[cf.
(i-a)]
given
by
mapping
T
Π
×
Z
(σ,
z)
→
(σ,
z
·
φ(σ))
∈
T
Π
×
Z.
Next,
let
out
def
us
observe
that
the
composite
H
C
=
N
T
out
T
(C)
→
T
Π
Π
T
H
H
G
k
X
is
surjective,
with
kernel
equal
to
I
C
.
Thus,
it
follows
from
Fact
(1)
in
the
proof
of
Corollary
4.18
that
this
composite
H
C
G
k
X
admits
a
section,
which
determines
an
isomorphism
out
∼
out
(
T
Π
T
I)
G
k
X
−→
T
Π
T
H.
Let
us
fix
such
a
section.
Next,
observe
that
it
follows
from
(i-
b),
(i-c)
that
the
above
automorphism
α
I
is
compatible
with
the
out
action
of
G
k
X
on
T
Π
T
I
determined
by
the
fixed
section
of
H
C
G
k
X
.
Thus,
we
conclude
that
the
above
automorphism
out
out
α
I
of
T
Π
T
I
extends
to
an
automorphism
α
of
T
Π
T
H
COMBINATORIAL
ANABELIAN
TOPICS
II
161
that
preserves
and
induces
the
identity
automorphism
on
the
image
of
the
fixed
section
of
H
C
G
k
X
.
Now
let
us
observe
that
it
is
immediate
that
out
α
I
,
hence
also
α,
does
not
preserve
T
Π
⊆
T
Π
T
I,
as
desired.
Let
us
also
observe
that
since
C
⊆
T
Π
is
con-
tained
in
the
kernel
of
φ,
it
follows
from
(i-a)
that
α
I
pre-
serves
and
induces
the
identity
automorphism
on
the
cuspidal
out
I-decomposition
subgroup
I
C
⊆
T
Π
T
I.
In
particular,
we
conclude
immediately
that
out
(i-d)
the
automorphism
α
of
T
Π
T
H
preserves
and
induces
the
identity
automorphism
on
H
C
.
(ii)
Next,
we
consider
the
case
of
a
singular
stable
log
curve
[i.e.,
a
stable
log
curve
that
is
not
smooth].
In
the
situation
of
(i),
let
W
log
be
a
stable
log
curve
over
(Spec
k
X
)
log
such
that
•
W
log
has
precisely
two
irreducible
components
each
of
which
is
a
tripod,
•
W
log
has
a
single
node,
and,
moreover,
log
•
the
natural
action
of
G
log
k
X
on
the
dual
semi-graph
of
W
is
trivial.
[Thus,
W
log
is
of
type
(0,
4).]
Then,
by
taking
“
W
H”
to
be
G
log
k
X
,
out
we
obtain
a
profinite
group
W
Π
W
H
[cf.
the
situation
and
notational
conventions
of
Corollary
4.18].
In
the
remainder
of
the
discussion
of
the
present
(ii),
out
we
construct
an
automorphism
of
W
Π
W
H
that
out
does
not
preserve
the
closed
subgroup
W
Π
⊆
W
Π
W
H.
Write
v
1
,
v
2
for
the
distinct
two
irreducible
components
of
out
W
log
.
Let
V
1
,
V
2
⊆
W
Π
⊆
W
Π
W
H
be
verticial
subgroups
of
W
out
def
Π
W
H
associated
to
v
1
,
v
2
such
that
N
=
V
1
∩
V
2
=
{1},
which
thus
[cf.
[NodNon],
Lemma
1.9,
(i)]
implies
that
N
is
a
out
nodal
subgroup
of
W
Π
W
H.
For
each
i
∈
{1,
2},
write
def
H
V
i
=
N
W
out
W
(V
i
),
Π
H
def
H
N
=
N
W
out
W
(N
).
Π
H
Then
one
verifies
immediately
[cf.
[CmbGC],
Proposition
1.2,
(ii)]
that
162
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(ii-a)
there
exists
a
commutative
diagram
of
profinite
groups
N
⊆
↓
C
V
i
⊆
↓
⊆
T
Π
⊆
T
H
V
i
⊇
H
N
↓
↓
out
Π
T
H
⊇
H
C
—
where
the
horizontal
arrows
are
the
natural
inclusions,
and
the
vertical
arrows
are
isomorphisms.
Moreover,
it
follows
immediately
from
a
similar
argument
to
the
argument
applied
in
the
proof
of
[CmbCsp],
Proposition
1.5,
(iii)
[i.e.,
in
essence,
from
the
evident
analogue
for
semi-
graphs
of
anabelioids
of
the
“van
Kampen
Theorem”
in
ele-
mentary
algebraic
topology],
that
(ii-b)
the
natural
inclusions
out
H
V
1
→
W
Π
W
H
←
H
V
2
determine
an
isomorphism
∼
out
lim
(H
V
1
←
H
N
→
H
V
2
)
−→
W
Π
W
H
−→
—
where
the
inductive
limit
is
taken
in
the
category
of
profinite
groups
—
which
restricts
to
an
isomorphism
of
closed
subgroups
∼
lim
(V
1
←
N
→
V
2
)
−→
W
Π
−→
—
where
the
inductive
limit
is
taken
in
the
category
of
profinite
groups.
On
the
other
hand,
it
follows
from
(i-d)
and
(ii-a)
that,
for
each
i
∈
{1,
2},
α
determines
an
automorphism
β
i
of
H
V
i
that
•
does
not
preserve
V
i
⊆
H
V
i
but
•
preserves
and
induces
the
identity
automorphism
on
the
closed
subgroup
H
N
⊆
H
V
i
.
Thus,
by
(ii-b),
β
1
and
β
2
determine
an
automorphism
γ
of
out
W
Π
W
H
that
does
not
preserve
the
closed
subgroup
W
Π
⊆
W
Π
W
H,
as
desired.
out
Lemma
4.19
(An
explicit
description
of
a
power
of
the
cyclo-
tomic
character).
Let
J
be
a
profinite
group,
ρ
J
:
J
→
Aut(G
0
)
a
continuous
homomorphism,
and
I
⊆
J
a
normal
closed
subgroup
of
J
such
that
either
COMBINATORIAL
ANABELIAN
TOPICS
II
163
ρ
J
(a)
the
composite
I
→
J
→
Aut(G
0
)
is
of
SNN-type
[cf.
[NodNon],
Definition
2.4,
(iii)],
or
(b)
I
=
{1}.
Write
def
out
def
out
Π
I
=
Π
G
0
I
⊆
Π
J
=
Π
G
0
J
[cf.
the
discussion
entitled
“Topological
groups”
in
[CbTpI],
§0].
Thus,
we
have
a
commutative
diagram
of
profinite
groups
1
−−−→
Π
G
0
−−−→
Π
I
−−−→
I
−−−→
1
⏐
⏐
⏐
⏐
1
−−−→
Π
G
0
−−−→
Π
J
−−−→
J
−−−→
1
—
where
the
horizontal
sequences
are
exact,
and
the
vertical
arrows
are
the
natural
inclusions.
Write
G
0
→
G
0
for
the
universal
covering
of
G
0
corresponding
to
Π
G
0
.
Let
e
0
be
a
node
of
G
0
.
Write
Π
e
0
⊆
Π
G
0
for
the
nodal
subgroup
associated
to
e
0
.
Write
Π
e
0
,J
⊆
Π
J
for
the
[necessarily
open]
subgroup
consisting
of
the
elements
σ
∈
Π
J
such
that
the
natural
action
of
σ
on
the
underlying
semi-graph
G
0
of
G
0
stabilizes
the
two
branches
of
the
node
e
0
(G
0
)
of
G
0
determined
by
e
0
.
Then
the
following
hold:
(i)
Let
N
be
a
positive
integer
and
γ
an
element
of
Π
J
.
Then
there
exists
a
collection
of
data
as
follows
•
a
normal
open
subgroup
H
⊆
Π
J
of
Π
J
,
•
a
positive
integer
m,
•
verticial
subgroups
Π
v
0
,
Π
v
1
,
.
.
.
,
Π
v
m−1
⊆
Π
G
0
of
Π
G
0
asso-
ciated
to
vertices
v
0
,
v
1
,
.
.
.
,
v
m−1
of
G
0
,
respectively,
and
•
nodal
subgroups
Π
e
1
,
.
.
.
,
Π
e
m
⊆
Π
G
0
of
Π
G
0
associated
to
nodes
e
1
,
.
.
.
,
e
m
of
G
0
,
respectively,
such
that
if
we
write
def
D
e
j
=
N
Π
I
(Π
e
j
)
for
each
j
∈
{0,
1,
.
.
.
,
m}
[cf.
[NodNon],
Definition
2.2,
(iii)],
then
(1)
the
inclusions
Π
e
0
⊆
Π
v
0
,
Π
e
m
⊆
Π
v
m−1
[which
imply
that
e
0
,
e
m
abut
to
v
0
,
v
m−1
,
respectively
—
cf.
[NodNon],
Lemma
1.7]
hold,
(2)
if
m
≥
2,
then,
for
every
j
∈
{1,
.
.
.
,
m
−
1},
the
inclusion
Π
e
j
⊆
Π
v
j−1
∩
Π
v
j
[which
implies
that
e
j
abuts
to
v
j−1
and
v
j
—
cf.
[NodNon],
Lemma
1.7]
holds,
(3)
the
quotient
D
e
0
D
e
0
⊗
Z
Σ0
(
Z
Σ
0
/N
Z
Σ
0
)
164
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
(
Z
Σ
0
/N
Z
Σ
0
)
×
(
Z
Σ
0
/N
Z
Σ
0
)
if
(a)
is
satisfied
Z
Σ
0
/N
Z
Σ
0
if
(b)
is
satisfied
[cf.
[CmbGC],
Remark
1.1.3;
[NodNon],
Lemma
2.5,
(i);
[NodNon],
Remark
2.7.1]
of
D
e
0
factors
through
the
quo-
tient
of
D
e
0
determined
by
the
composite
D
e
0
→
Π
J
Π
J
/H,
and,
moreover,
(4)
the
image
of
D
e
m
⊆
Π
J
in
Π
J
/H
coincides
with
the
im-
age
of
γ
·
D
e
0
·
γ
−1
⊆
Π
J
in
Π
J
/H.
For
each
j
∈
{0,
1,
.
.
.
,
m
−
1},
write
∼
=
def
def
D
v
j
=
N
Π
I
(Π
v
j
)
⊇
I
v
j
=
Z
Π
I
(Π
v
j
)
=
Z(D
v
j
)
[cf.
[NodNon],
Definition
2.2,
(i);
[NodNon],
Lemma
2.5,
(i);
[NodNon],
Remark
2.7.1;
[CmbGC],
Remark
1.1.3];
b
j,j
,
b
j+1,j
for
the
respective
branches
of
the
nodes
e
j
,
e
j+1
that
abut
to
the
vertex
v
j
determined
by
the
inclusions
Π
e
j
⊆
Π
v
j
,
Π
e
j+1
⊆
Π
v
j
[cf.
(1),
(2)].
Thus,
for
j
∈
{0,
1,
.
.
.
,
m}
and
s
∈
{0,
1,
.
.
.
,
m
−
1}
such
that
s
∈
{j
−
1,
j},
it
follows
from
[NodNon],
Remark
2.7.1,
that
we
have
natural
inclusions
I
v
s
⊆
D
e
j
⊆
D
v
s
∪
∪
Π
e
j
⊆
Π
v
s
,
which
determine
a
commutative
diagram
of
profinite
groups
D
e
j
/I
v
s
−−−→
D
v
s
/I
v
s
⏐
⏐
⏐
⏐
Π
e
j
−−−→
Π
v
s
—
where
the
horizontal
arrows
are
the
natural
inclusions,
and
the
vertical
arrows
are
isomorphisms.
(ii)
In
the
situation
of
(i),
by
applying
the
construction
of
“Λ
v
”
of
[CbTpI],
Definition
3.8,
(i),
to
the
collection
of
data
consisting
of
•
the
profinite
group
D
v
s
/I
v
s
and
•
the
various
images
in
D
v
s
/I
v
s
,
by
the
right-hand
vertical
∼
isomorphism
Π
v
s
→
D
v
s
/I
v
s
of
the
final
display
of
(i),
of
the
edge-like
subgroups
of
Π
G
0
contained
in
Π
v
s
,
one
may
construct
a
cyclotome
Λ(D
v
s
/I
v
s
).
Moreover,
by
applying
the
construction
of
“syn
b
”
of
[CbTpI],
Corollary
3.9,
(v),
to
the
collection
of
data
consisting
of
•
the
profinite
groups
D
e
j
/I
v
s
,
D
v
s
/I
v
s
,
COMBINATORIAL
ANABELIAN
TOPICS
II
165
•
the
various
images
in
D
v
s
/I
v
s
,
by
the
right-hand
vertical
∼
isomorphism
Π
v
s
→
D
v
s
/I
v
s
of
the
final
display
of
(i),
of
the
edge-like
subgroups
of
Π
G
0
contained
in
Π
v
s
,
and
•
the
upper
horizontal
arrow
D
e
j
/I
v
s
→
D
v
s
/I
v
s
of
the
final
display
of
(i)
[i.e.,
that
corresponds
the
branch
b
j,s
],
one
may
construct
an
isomorphism
∼
syn
b
j,s
:
D
e
j
/I
v
s
−→
Λ(D
v
s
/I
v
s
).
Write
def
M
v
s
=
M
e
j
I
v
s
⊗
Z
Σ0
Λ(D
v
s
/I
v
s
)
if
(a)
is
satisfied
if
(b)
is
satisfied
Λ(D
v
s
/I
v
s
)
det(D
e
j
)
if
(a)
is
satisfied
def
=
D
e
j
/I
v
s
if
(b)
is
satisfied
—
where
the
“det”
is
taken
with
respect
to
the
structure
of
free
Z
Σ
0
-module
of
finite
rank
of
the
profinite
group
D
e
j
;
we
observe
that
if
condition
(a)
is
satisfied,
then
the
exact
sequence
of
free
Z
Σ
0
-modules
of
finite
rank
1
−→
I
v
s
−→
D
e
j
−→
D
e
j
/I
v
s
−→
1
yields
a
natural
identification
M
e
j
=
I
v
s
⊗
Z
Σ0
(D
e
j
/I
v
s
)
of
Z
Σ
0
-modules
[cf.
[CmbGC],
Remark
1.1.3;
[NodNon],
Lemma
2.5,
(i);
[NodNon],
Remark
2.7.1];
we
observe
that
if
condition
(b)
is
satisfied,
then
since
I
v
s
=
{1},
we
have
a
natural
isomor-
∼
phism
D
e
j
→
D
e
j
/I
v
s
=
M
e
j
.
If
condition
(a)
is
satisfied,
then
let
us
write
M
∼
syn
b
j,s
:
M
e
j
=
I
v
s
⊗
Z
Σ0
(D
e
j
/I
v
s
)
−→
I
v
s
⊗
Z
Σ0
Λ(D
v
s
/I
v
s
)
=
M
v
s
for
the
isomorphism
determined
by
the
above
isomorphism
syn
b
j,s
.
If
condition
(b)
is
satisfied,
then
let
us
write
M
∼
def
syn
b
j,s
=
syn
b
j,s
:
M
e
j
=
D
e
j
/I
v
s
−→
Λ(D
v
s
/I
v
s
)
=
M
v
s
.
def
(iii)
In
the
situation
of
(ii),
write
n
0
=
2
(respectively,
1)
if
condi-
tion
(a)
(respectively,
(b))
is
satisfied.
Write
Φ
N
(γ)
∈
Aut(M
e
0
⊗
Z
Σ0
(
Z
Σ
0
/N
Z
Σ
0
))
=
(
Z
Σ
0
/N
Z
Σ
0
)
∗
for
the
automorphism
of
the
free
Z
Σ
0
-module
[of
rank
one]
M
e
0
⊗
Z
Σ0
(
Z
Σ
0
/N
Z
Σ
0
)
obtained
by
forming
the
composite
of
the
isomorphism
∼
M
e
0
⊗
Z
Σ0
(
Z
Σ
0
/N
Z
Σ
0
)
−→
M
e
m
⊗
Z
Σ0
(
Z
Σ
0
/N
Z
Σ
0
)
166
YUICHIRO
HOSHI
AND
SHINICHI
MOCHIZUKI
determined
by
conjugating
by
γ
∈
Π
J
[cf.
conditions
(3),
(4)
in
(i)]
with
the
isomorphism
∼
M
e
m
⊗
Z
Σ0
(
Z
Σ
0
/N
Z
Σ
0
)
−→
M
e
0
⊗
Z
Σ0
(
Z
Σ
0
/N
Z
Σ
0
)
determined
by
the
inverse
of
the
composite
M
syn
−1
b
1,0
M
syn
b
0,0
∼
∼
M
syn
−1
b
2,1
M
syn
b
1,1
∼
∼
M
e
0
−→
M
v
0
−→
M
e
1
−→
M
v
1
−→
M
syn
−1
bm,m−1
M
syn
bm−1,m−1
∼
∼
...
−→
M
v
m−1
−→
M
e
m
.
Suppose
that
γ
∈
Π
e
0
,J
.
Then
the
image
of
γ
by
the
composite
⊗2n
0
χ
G
ρ
J
0
Π
J
−→
J
−→
Aut(G
0
)
−→
(
Z
Σ
0
)
∗
−→
(
Z
Σ
0
/N
Z
Σ
0
)
∗
[cf.
[CbTpI],
Definition
3.8,
(ii)]
coincides
with
Φ
N
(γ)
2
∈
(
Z
Σ
0
/N
Z
Σ
0
)
∗
.
(iv)
Let
ρ
:
Π
J
→
(
Z
Σ
0
)
∗
be
a
character
[i.e.,
a
continuous
homo-
morphism]
and
n
ρ
a
positive
integer
divisible
by
2[Π
J
:
Π
e
0
,J
].
Suppose
that,
for
each
positive
integer
N
and
each
γ
∈
Π
e
0
,J
,
the
image
of
ρ(γ
)
2
∈
(
Z
Σ
0
)
∗
in
(
Z
Σ
0
/N
Z
Σ
0
)
∗
coincides
with
Φ
N
(γ
)
2
∈
(
Z
Σ
0
/N
Z
Σ
0
)
∗
[cf.
(iii)].
Then
the
n
ρ
-th
power
of
the
character
ρ
coincides
with
the
n
ρ
-th
power
of
the
character
obtained
by
forming
the
composite
ρ
⊗n
0
χ
G
J
0
Π
J
−→
J
−→
Aut(G
0
)
−→
(
Z
Σ
0
)
∗
[cf.
(iii)].
Proof.
Assertions
(i),
(ii)
follow
immediately
from
the
various
defini-
tions
involved.
Next,
we
verify
assertion
(iii).
Let
us
first
observe
that
it
follows
immediately
from
the
various
definitions
involved
that
there
exist
δ
∈
Π
G
0
⊆
Π
e
0
,J
and
∈
N
Π
J
(Π
e
0
)
∩
Π
e
0
,J
(⊆
N
Π
J
(D
e
0
)
∩
Π
e
0
,J
)
such
that
γ
=
δ
·
.
Now
one
verifies
immediately
from
[CbTpI],
Corol-
lary
3.9,
(ii),
(v);
[CbTpI],
Corollary
5.9,
(ii),
that
the
action
of
on
M
e
0
by
conjugation
is
given
by
multiplication
by
χ
G
0
(
)
n
0
.
Moreover,
let
us
observe
that
one
verifies
easily
that
the
collection
of
data
of
as-
sertion
(i)
[i.e.,
associated
to
γ]
satisfies
conditions
(1),
(2),
(3),
(4)
in
assertion
(i)
in
the
case
where
we
take
“γ”
to
be
δ.
Also,
let
us
observe
that
the
image
of
δ
by
the
composite
ρ
J
⊗n
0
χ
G
0
Π
J
−→
J
−→
Aut(G
0
)
−→
(
Z
Σ
0
)
∗
−→
(
Z
Σ
0
/N
Z
Σ
0
)
∗
is
trivial.
Thus,
assertion
(iii)
follows
immediately
from
[CbTpI],
Corol-
lary
3.9,
(ii),
(v),
(vi).
This
completes
the
proof
of
assertion
(iii).
As-
sertion
(iv)
is
a
formal
consequence
of
assertion
(iii).
This
completes
the
proof
of
Lemma
4.19.
COMBINATORIAL
ANABELIAN
TOPICS
II
167
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